B. Infer, distinguish and revise

MMP.7 - Construct a Probability Model of the Recorded Data

Type: Method pattern Status: Stable Normativity: Normative within the stated use

MMP.7:1 - Problem frame

Use this pattern when inference depends on how events, responses or quantities become records, and that procedure has not yet been expressed in the probability model. A feedback log may contain successes more often than failures. A timed trial can end before its event occurs. Several readings can share one calibration error. In each case, fitting a familiar distribution to the visible numbers can answer a different question from the one you intended.

Start with one possible event and follow what the observing procedure would record. Include the possibility that it leaves no record, reports an interval or shares an influence with another observation. Repeat for a contrasting event. These cases reveal what the mathematical outcome must contain before you choose its distribution.

The result is a probability law for the recorded outcome under stated assumptions, together with its relation to the quantity being inferred. It can supply a likelihood, a distribution of future records or a reason the intended inference remains ambiguous. This pattern develops probabilistic formulation within mathematical modeling. The subject practice supplies the meaning of the event, the observation procedure and plausible relations among quantities.

You need conditional probability and sums over alternatives; continuous cases also use densities and integration. A collaborator can supply those operations when you can describe the observation procedure and interpret the returned law. If an existing model already represents that procedure and answers the question, use it. When only compatible ranges are needed, C.16.IR can provide a sufficient answer without probabilities.

MMP.7:2 - Problem

The distribution of a subject property and the distribution of its records can differ. Selection changes which cases appear. Coarsening combines several possible values into one report. A common influence makes observations dependent. These transformations remain part of the inference even when a dataset presents every row in the same format.

A formula such as independent errors around a predicted value already makes choices about those transformations. If the choices are left implicit, more data and more accurate computation can reinforce a mistaken interpretation.

The difficulty is to construct the law of the observations from the modeled subject and its recording procedure, preserving the dependencies that matter to the question.

MMP.7:3 - Forces

MMP.7:4 - Solution

Construct the possible recorded outcomes from the observation procedure. Combine the subject and recording laws, remove the unobserved alternatives by the appropriate probability operation, and inspect what the resulting law permits you to infer. Return to the procedure or assumptions when its output does not answer the working question.

MMP.7:4.1 - Choose the target and the recorded outcome separately

State what the answer concerns: a rate in a population, a property before measurement, a future response, or another quantity selected by the work. Specify the population, conditions and time range when they change its meaning. Use C.16 for the characteristic being measured and C.16.MR for the relation from the property to an indication.

Describe one complete outcome of the observing procedure. It may contain a value and an inclusion flag, a duration and a timeout flag, or several related readings. The mathematical outcome space must distinguish every report the procedure can produce that affects the inference.

State what the observation plan fixes. Following a known cohort produces information about excluded cases that a sample drawn only from submitted reports may lack. Stopping after a specified time, after a specified number of records, or after an event can produce different data laws. Recover the actual plan before treating any count as fixed.

MMP.7:4.2 - Construct the joint law from the modeled dependencies

Introduce variables for the quantities used by that procedure. Explain their domains and meanings before assigning distributions. Let Z denote an underlying event or value and O its recorded outcome. Parameters theta describe quantities held fixed in the proposed probability model. When these laws are represented by probability masses or by densities under an appropriate reference measure, write the subject law as p_theta(z) and the conditional recording law as k_theta(o given z). Their joint expression is:

p_theta(z,o) = p_theta(z) k_theta(o given z).

Each factor needs an interpretation. The first describes variation in the subject under the stated conditions; the second describes how the procedure records it. A deterministic recorder assigns probability one to its specified output and zero to the other outputs. This accommodates rounding and threshold reports as well as random response or selection.

Use a sequence of conditional laws when more stages matter. Multiplication follows the chain rule. Omitting a variable from a conditional law asserts that, given the retained variables, it does not change that law. Make that assumption from the modeled relation; separate rows in a file provide no independence argument.

Keep an unknown fixed parameter as unknown. Give it a probability distribution only when that additional modeling choice is justified for the intended inference. A shared but unknown calibration offset can remain a parameter in a joint likelihood. A distribution over possible offsets supports a different, explicitly extended model.

MMP.7:4.3 - Obtain the law for what was actually recorded

The general operation averages the chance of an observed event over the underlying cases. Let K_theta(B given z) be the chance that the report falls in a set B, given underlying value z. Then:

P_theta(O in B) = integral K_theta(B given z) P_theta(dz).

Here P_theta(dz) means averaging with the probability law of Z: a weighted sum for discrete cases or an integral for continuous ones. A deterministic recorder O=g(Z) has K equal to one when g(z) lies in B and zero otherwise. This constructs its output law even when the joint pair (Z,O) has no ordinary joint density, as with O=Z for a continuously varying Z.

When the masses or densities used in :4.2 are available, the same averaging operation gives the law of a particular report. For discrete unobserved alternatives, sum:

p_theta(o) = sum_z p_theta(z) k_theta(o given z).

For continuous alternatives, integrate the product of the subject density and the recording factor. A report produced exactly when Z lies in a fixed set A has recording factor one inside A and zero outside; its probability reduces to the integral of the density over A. If the procedure chooses which set to report, retain that choice in k_theta(A given z).

For example, let Z be equally likely to be 0 or 1. A truthful recorder reports {0,1} always when Z=0 and with probability 1/2 when Z=1; otherwise it reports {1}. The probability of receiving {0,1} is 1/2 + (1/2)(1/2) = 3/4, although the probability that Z lies in {0,1} is one. The recording factor makes the difference.

For an individually observed continuous value, use a density with respect to the stated measurement convention. A point density and the probability of an interval have different meanings.

When inclusion in the dataset is itself a condition of sampling, retain its normalization. If Z has density or mass p_theta(z), and s_theta(z) is its probability of inclusion, the included-case law is:

p_theta(z given included) = p_theta(z) s_theta(z) / P_theta(included).

The denominator is obtained by summing or integrating the numerator over all admitted z and must be positive. If it depends on theta, dropping it changes the inference. When the counts or identities of excluded cases are also observed, include that information in the joint outcome instead of silently discarding it by conditioning. Section :5.1 shows the change.

Keep shared influences shared during elimination. For observations conditionally independent given an unknown B, integrating one joint product over B generally differs from multiplying separately integrated factors. The latter construction assigns a fresh B to each observation. Use it only when that is the observing arrangement.

MMP.7:4.4 - Connect the law to inference and prediction

When the observation laws have a common probability-mass or density representation, insert the recorded outcome o into p_theta(o). As a function of theta, this gives a likelihood, up to a factor independent of theta. It need not sum or integrate to one over theta. Estimation or a posterior distribution requires the chosen inferential method and its assumptions; the observation law is the input to that work.

Before drawing an inference, check whether the recording rule admits the received report for any parameter value. A continuous reading can be admitted even though its single-point probability is zero; determine admissibility from the modeled observation mechanism and the cases it permits. If no admitted case produces the report, return the conflict and locate which assumptions or recording steps need reconsideration, using C.16.IR:4.4. For example, a fixed signal with one fixed additive offset and one unchanged threshold must produce identical bits on repetition. A mixed sequence contradicts that joint account. It cannot be repaired by fitting a different signal within the same family.

Identify how the requested quantity depends on theta or on a future outcome. Two parameter settings can induce the same law for every possible record while assigning different values to the target. Constructing such a pair shows that this observation model cannot identify that distinction. C.16.IR supplies the corresponding compatible-case reasoning; numerical fitting alone cannot resolve it.

For a future record, specify whether its recording procedure is the same. For the underlying population quantity, return through the subject law rather than interpreting a selected-case rate as the population rate. A proposed intervention requires its changed relations under C.28; changing a predictor value in a fitted association is insufficient when the intervention changes how the data arise.

MMP.7:4.5 - Test a consequence and revise the construction

Check normalization and a small case that follows the procedure. Enumerate a finite outcome space or generate subject cases and pass them through the recorder. Compare that construction with the probabilities or summaries derived from the observation law. C.29.2 supplies a computational construction when enumeration or integration needs further work.

When the inclusion rule, timeout, shared calibration or receiving question actually changes, revise the affected relation and carry its consequence through the calculation. Compare a plausible alternative condition when the comparison can change the intended use or returned claim; this can expose the observation mechanism beyond one fixed formula. An already sufficient construction under unchanged conditions needs no invented variation.

A simulation agreeing with the formula checks their agreement under the modeled assumptions. An available observation can challenge those assumptions; selected domain assurance determines which empirical comparison is worth performing. C.11.DUA helps choose between further observation, a conditional answer and acting with remaining uncertainty. Preserve a sufficient result without demanding another dataset merely because an influence remains unknown.

MMP.7:5 - Archetypal Grounding

MMP.7:5.1 - Infer a success rate from a selectively submitted log

A team asks what fraction of attempts succeed. In a proposed model, each attempt succeeds with probability p. Every success is logged; each failure is logged independently with probability 1/4. Initially the team has a fixed-size sample of independently drawn log entries, with no information about how many attempts produced the source log.

The event variable Y is success or failure. The recording flag R says whether the attempt enters the log. Their joint probabilities are:

The included-case success probability is q = p / [p + (1-p)/4] = 4p/(1+3p). If the observed fraction of successes is 1/2, the likelihood estimate of q is 1/2, and transforming it gives p_hat = q_hat/(4-3q_hat) = 1/5. Sampling uncertainty remains; this calculation corrects which rate is being estimated.

The first useful result is the distinction between a 50% rate among reports and the estimated 20% rate among attempts under the supplied reporting assumptions. If the failure-reporting probability is unknown, several combinations of that probability and p can produce the same q. The log alone then leaves the population rate unresolved.

Now the procedure changes: a register names a fixed cohort of N attempts and links each submitted report to its attempt. Model those attempts as independent, each with the same success probability p, retaining the stated reporting rule. Since every success is reported, an unreported attempt is a failure. If there are k success reports, the likelihood for p is proportional to p^k (1-p)^(N-k); the failure-reporting factors do not depend on p. The estimate becomes k/N. Conditioning only on reported entries would throw away information the revised procedure provides.

This is a change in the team’s observing method. A.3.2.DE explains the changed recording rule; C.32.MWA.PP connects that contribution to actual performers and support. Whether to introduce it depends on what resolving the population rate would change in the team’s work.

MMP.7:5.2 - Preserve a common influence across readings

Two sensors measure quantities x1 and x2 with one shared calibration offset b. Their readings are Y1=x1+b+E1 and Y2=x2+b+E2, with independent zero-mean errors of variance sigma squared. Begin by retaining b as a common parameter. The joint conditional density factors given b; each factor uses that same value.

For the difference, Y1-Y2=x1-x2+E1-E2: the offset cancels. Its error variance is 2 sigma^2. A measurement of the difference can therefore be useful while either absolute value remains uncertain.

For repeated measurements of one x, suppose an additional justified model describes the common offset as a zero-mean random variable B with variance tau squared, independent of the errors. The average of n readings has variance tau^2 + sigma^2/n. Integrating a separate offset for every reading would incorrectly produce (tau^2+sigma^2)/n. Repetition reduces independent noise but leaves this common calibration contribution.

If the instrument is independently recalibrated before every reading, the arrangement changes. A separate-offset model can then be appropriate. The governing operation is to trace which influences are shared and preserve that sharing in the probability construction.

MMP.7:5.3 - Use a timed-out trial as an interval report

A test asks how long an event takes. Each independent trial is observed until its event or a fixed timeout c. Record both V=min(T,c) and a flag D indicating whether the event occurred before timeout. As an illustrative subject assumption, let T have exponential density lambda exp(-lambda t) for t at least zero, with lambda positive.

An event at time t before c contributes the density lambda exp(-lambda t). A timeout contributes P(T>=c)=exp(-lambda c), obtained by integrating the density over the unobserved tail. For m completed trials and total observed time S, including the timeout durations, the likelihood is proportional to lambda^m exp(-lambda S).

With completions at times 1 and 2 and one timeout at 4, S=7 and m=2. Maximizing this illustrative likelihood gives lambda_hat=2/7. Treating the timeout as a third event instead gives 3/7; dropping it gives 2/3. The flag determines which operation is correct.

If only completed trials enter a database and neither the number nor identities of timed-out trials are available, the observed-time density instead conditions on completion: divide the event density by 1-exp(-lambda c) on the interval before c. A changed recording rule changes the model even when the stored times look the same. The exponential assumption is dispensable: a different duration law supplies its own event density and tail probability.

MMP.7:6 - Bias-Annotation

The visible dataset invites treating its rows as the whole observation procedure. Begin from how a row, absence or interval report is produced. A second temptation is to add one independent error to every row; trace shared influences before factorizing. More detailed modeling can also conceal missing knowledge, so retain uncertainty in the recording mechanism when the available information does not determine it.

MMP.7:7 - Conformance Checklist

• Can a reader identify the target and all possible reports that affect the inference?
• Do the subject and recording factors describe the stated procedure, including what it fixes and what it reveals?
• Are unobserved alternatives removed by a justified sum, integral or conditioning operation?
• Does the joint law preserve common influences and any information about excluded cases?
• Is the first result interpreted as a likelihood, estimate, prediction, bound or unresolved distinction with its respective conditions?
• Can the formulation be changed when the observing procedure or receiving question changes?

MMP.7:8 - Common Anti-Patterns and How to Avoid Them

MMP.7:9 - Consequences

The constructed law allows computation to answer the intended observation question and can expose an ambiguity before expensive fitting. A changed reporting procedure becomes a model change that can be analyzed. The work may also show that the target needs assumptions or information absent from the records; a narrower conditional answer can remain useful.

MMP.7:10 - Architectural Rationale

The observation procedure connects the subject to the data used in inference. Keeping that connection explicit makes deterministic coarsening, random selection and shared uncertainty instances of one construction. It also separates modeling choices from the later choice of an inference algorithm.

The factors are chosen for the procedure and question. Their order as a probability factorization does not establish a causal direction in the represented world. Several factorizations can describe one joint law; the subject account and intervention question determine any causal interpretation.

This method uses C.16’s measurement and resolvability work while supplying the probability operations those patterns leave to statistical modeling. Its examples require different transformations: conditioning after selection, joint elimination and tail integration. The transferable operation survives replacing the logged activity, sensor or timed event.

MMP.7:11 - SoTA-Echoing

Gelman, Vehtari and McElreath, Statistical Workflow (2025), sections 1.1-1.7, emphasize measurement, assumptions, shared information and the distinction between model parameters and inferential targets. Adopt the connection of those decisions to model construction and revision. Their comparison of Bayesian and other workflows supports leaving the inference method explicit; it does not select one estimator for every observation law.

Rubin, Inference and missing data (1976), is a historical foundation for specifying when a missing-data mechanism can be ignored. Adopt the requirement to establish the applicable conditions rather than assuming that absence is harmless. Its qualifications depend on the inferential method. The examples here derive their recording laws directly and require no blanket ignorability claim.

The Stan User’s Guide 2.39 treatment of truncation and censoring provides executable constructions for restricted observations and tail reports. Adapt those probability operations to :4.3 and :5.3. The guide’s programming and inference conventions are useful implementations, not prerequisites of the method. A direct finite calculation or another suitable implementation can supply the same law.

The three demonstrations are elementary constructions for this pattern. They explain the modeling operations under stated assumptions; they are not empirical reports about the activities or devices used as examples.

MMP.7:12 - Relations

C.16.MR constructs the relation from a sought property to its indication; this pattern makes its probability law usable for inference. C.16.IR identifies what the resulting observations can resolve. C.29.2 constructs a computation for marginalization, estimation or prediction when needed, and C.29.3 addresses its realization. C.28 governs a causal use of the result. MMP.8 uses the observation model to determine the information available to a decision. B.5.MPC.R coordinates a repair spanning subject interpretation, mathematical formulation and computation. MMP.22 uses a mathematical transformation to construct a changed observing rule; C.32.MWA.PP compares its practical conditions and A.3.1.MV retains a consequential Method variant.

MMP.7:End

MMP.12 - Formulate an Inverse Problem and Choose Whether and How to Use Regularization

Type: Method pattern Status: Stable Normativity: Normative within the stated use

MMP.12:1 - Problem frame

Use this pattern when you can predict records from a proposed state or model, but recovering the needed unknown from the records is ambiguous or too sensitive to their errors. A mixture can reveal its total while scarcely distinguishing its components. Accumulated activity can be known much more reliably than its instantaneous rate. You need to decide what can be recovered and what additional structure a usable reconstruction would impose.

Begin with the quantity the next use needs. Write how a candidate unknown would produce the recorded result, then vary the unknown in a direction that changes that quantity. If the records remain unchanged, that distinction is unidentified. If they change only slightly, calculate how their uncertainty affects recovery. Introduce regularization only when the remaining task warrants its extra assumptions.

Regularization constructs a controlled reconstruction by restricting candidates or discouraging selected variations. Its practical gain is a usable answer whose dependence on that restriction is visible. It can suppress error amplification while also suppressing real detail. The result may be a conditional estimate, a sufficient target bound, or a reason the requested reconstruction needs another contribution.

This is the inverse-formulation branch of mathematical modeling. It constructs the reconstruction problem and the added structure that shapes its answer. Statistical inference supplies probability claims when needed; computational methods obtain solutions to the stated problem. The examples use linear equations, norms and elementary differentiation. More difficult operators require suitable mathematical preparation or a collaborator who can supply their inverse and stability analysis. A person or AI participant still needs the subject grounds for the forward relation and the proposed restriction.

Use an adequate direct inverse when its propagated error is acceptable. If an available range or comparison already settles the receiving question, return it through C.16.IR without reconstructing every unknown. Solver implementation alone calls for the corresponding computational method.

MMP.12:2 - Problem

Matching the data can select the wrong kind of answer. Several unknowns may produce the same records, or the best fit may follow noise in a direction the observations barely constrain. More accurate solution of those fitting equations can make the reconstruction more extreme.

Additional structure can help, but it changes the grounds of the answer. A small norm favors small unknowns in a chosen representation. A smoothness penalty favors slow variation. A learned penalty favors features represented by its training and construction. Each can remove a feature the receiving question needs.

The problem is to formulate recovery so that the required target, observation relation, error account and added selection are distinguishable, and then to determine which improvement and loss the regularization produces.

MMP.12:3 - Forces

MMP.12:4 - Solution

Construct the forward relation for the requested target, locate the consequential loss of distinction or amplification, and choose the smallest justified restriction that changes that difficulty. Formulate its strength together with the error account. Return the target with the dependence and loss introduced by that choice.

MMP.12:4.1 - Construct the forward relation from the modeled situation

Name the unknown x, the target q=T(x), and the records y. The unknown may be a parameter vector, a function or a collection of relations. State its domain and the units and scales used to compare changes. A target can be a total, a value at one time or a threshold; it need not be x itself.

Follow a candidate x through the modeled process and recording operation. Write the resulting ideal record as F(x,z), where z contains influential unknowns that are not the target. Keep known inputs fixed and retain shared unknowns across records. MMP.11 constructs a missing model family; C.16.MR supplies a missing measurement relation. If probabilities matter, MMP.7 constructs the recording law.

For an additive bounded-error account, one possible formulation is

y_delta = F(x,z) + e, with ||e||_Y <= delta.

Here delta bounds error in the chosen record norm. Use this form only when the recording procedure supports additive error. For an implicit forward relation, retain its equations and jointly unknown outputs rather than forcing it into a single-valued map. A likelihood from MMP.7 can instead supply the data discrepancy appropriate to a probabilistic account.

Include consequential uncertainty in calibration and in the forward approximation. A fixed but unknown offset remains an unknown; setting it to zero can make recovery appear better determined. A noise bound and a standard deviation have different meanings and support different conclusions.

MMP.12:4.2 - Locate the part that needs regularization

Use C.16.IR’s compatible cases or sufficient bounds to establish what the current records and premises resolve. Reuse that result. This pattern adds the construction of a recovery rule and analysis of the rule’s sensitivity.

For a linear relation y=A*x, a change h with A*h=0 is invisible to the records. If both x and x+h are admitted and T(x+h) differs from T(x), full recovery of that target needs another premise. If T is unchanged, the invisible direction may be irrelevant to the current use. For unknown influences, vary x and z jointly.

Next examine changes that are visible but weak. In finite dimensions, after meaningful scaling, singular values of A describe the response to orthogonal input directions. A small nonzero singular value sigma means that direct inversion multiplies the corresponding record error by 1/sigma. Estimate the effect on T, not merely on an unnecessarily detailed reconstruction. For nonlinear models, a derivative can reveal local weak directions; it does not establish global uniqueness or exclude another branch.

Distinguish poor conditioning from a discontinuous inverse. An invertible finite matrix has a continuous inverse, although its error amplification may be unacceptable. In a function-space problem, data changes tending to zero can produce target changes that do not tend to zero. The spaces and norms determine this claim; :5.3 gives an explicit example. Refining a finite discretization can expose progressively larger amplification.

If the supported target bound is already sufficient, stop. If the premises are inconsistent, return to their diagnosis through C.16.IR. A penalty cannot make an incompatible observation account true.

MMP.12:4.3 - Turn the added structure into a reconstruction problem

State why particular alternatives should be excluded or discouraged. A known nonnegative quantity can justify a hard domain restriction. A supported slowly varying response can justify discouraging rapid changes. A reference state can justify penalizing departure from it. When the structure is only a preference for selecting one nominal model, say so; the selected model then remains one conditional representative.

A useful variational formulation is

(x_lambda,z_lambda) in argmin over (x,z) in C of D(F(x,z),y_delta) + lambda*R(x,z).

C contains the hard conditions. D measures discrepancy in records. R is the regularizer: the quantity whose increase discourages a candidate. The parameter lambda controls its weight. Explain each term’s subject meaning and scaling. Adding squared errors with different units, or changing units without changing the weights, changes the problem.

Choose R by deriving which variations it penalizes. For a quadratic example,

R(x) = ||L*(x-x0)||^2.

The reference x0 and operator L determine the preference. L=I penalizes distance from x0; a difference operator penalizes changes between neighboring values. The latter permits constant shifts unless another condition fixes them. A sparsity penalty is appropriate only when concentrating the unknown in relatively few components fits the intended representation and subject question.

For a finite, unconstrained real linear problem with scaled squared discrepancy, differentiating the quadratic objective gives

(A^T*A + lambda*L^T*L)*x = A^T*y_delta + lambda*L^T*L*x0.

For lambda>0 this has a unique minimizer when the null spaces of A and L intersect only at zero. Thus adding a penalty does not by itself guarantee unique recovery. Constraints and nonlinear or nonconvex choices require their own existence, uniqueness and obtaining arguments.

An alternative is to minimize R subject to a justified discrepancy ceiling. This makes the acceptable fit explicit. It can agree with a penalized formulation for suitable parameters, but the correspondence must be established for the problem being used.

A learned regularizer is another way to construct R. Its training examples and training objective supply additional structure, whose relevance to the current subject and observation conditions must be justified. Compare it with the simpler available restriction on the same required target and with its training and obtaining costs included. A visually plausible or numerically precise reconstruction can still lose the feature the target asks for.

MMP.12:4.4 - Select strength from the required sensitivity and tolerated loss

Work out how the added structure changes recovery before choosing its numerical strength. For L=I, x0=0 and one nonzero singular direction, quadratic regularization replaces division by sigma with multiplication by

sigma/(sigma^2 + lambda).

This reduces noise amplification. With exact data, it also multiplies the true component by sigma^2/(sigma^2+lambda), shrinking it toward zero. A component invisible to A is selected through the regularizer, not recovered from the records.

Choose lambda using the error account and the receiving use. A bound on acceptable noise amplification, together with a bound on tolerated shrinkage, can determine an interval of useful values. Section :5.1 computes such a choice. A supported discrepancy level can instead guide a parameter search: compare each candidate’s residual with the level warranted by the observation and model errors. An empirical choice rule needs evidence appropriate to its own claim; fitting the available records best is not a general parameter-selection argument.

Evaluate both sides of the trade-off. Compare the changed target when the data are perturbed within their error account and when the reference, penalty or strength changes within its justified range. Return consequential dependence rather than hiding it behind one selected value. A few numerical trials can reveal a failure; they establish a bound only when an argument covers the claimed variation.

If no supported strength gives the needed sensitivity and tolerable loss, narrow the target or return the missing contribution. A stronger penalty can make outputs nearly constant while leaving them useless for the question.

MMP.12:4.5 - Separate recovery error from solving error

For a linear reconstruction rule H_lambda and exact data y, the triangle inequality separates two contributions:

||H_lambda*y_delta - x_target|| <= ||H_lambda*(y_delta-y)|| + ||H_lambda*y - x_target||.

The first is propagated data error; the second is the displacement caused by the reconstruction rule even with exact data. Choose x_target explicitly: an identified true unknown, a specified minimum-norm solution, or another admitted target. Those are different claims. Numerical approximation adds its own contribution, which MATH.20 and CMP.8 can bound.

For a fixed regularization strength, establish only the stability supported by the formulation. A unique minimizer in a general nonlinear problem is not automatically a quantitative stability bound. If the claim concerns recovery as noise tends to zero, specify how strength and any discretization change with that noise. Fixed-strength bias may persist. Classical regularization analysis supplies conditions for this limit; the existence of a penalty is insufficient.

Keep that limit separate from iterations of a solver converging at fixed data and strength. The solver can converge to the exact minimizer of a biased problem. Conversely, early stopping can itself be a regularization choice when its stopping rule has an appropriate noise-dependent justification. Additional iterations then need not improve the subject reconstruction.

For ordinary finite use, obtain only the error or settled distinction the receiver needs. A limiting theorem need not be proved anew when an applicable result and a sufficient finite bound are available.

MMP.12:4.6 - Return the useful target and the choice it depends on

Return T(x_lambda) with the forward relation, consequential error assumptions and added restriction needed to interpret it. State what remains unresolved and how much the target changes under the relevant data and regularization variations. Preserve a compatible range when it is needed alongside a nominal reconstruction. A penalty-selected point alone supplies neither a confidence interval nor a posterior distribution.

Regularization changes the grounds for selecting an answer; it creates no new observation from the old records. When a different premise, observation relation or target would remove the difficulty, name that contribution. Further computation helps only if the remaining problem is computational.

A person or AI can propose the formulation, calculate the example and compare alternatives. The needed subject relation and mathematical argument must still be available from a competent participant or established result. When they are missing, return the precise question they must answer. B.5.RR carries a changed premise through the reasoning; B.5.MPC.R helps coordinate a revision spanning subject, mathematical and computational contributions.

A methodological use can take the same result back to a working method: for example, replace an unstable instantaneous-rate target by a sufficient interval total, or revise how records are obtained when a needed distinction remains invisible. Such a change is selected for the original use, not made compulsory by the presence of an inverse problem.

MMP.12:5 - Archetypal Grounding

MMP.12:5.1 - Recover a split whose contrast is weakly observed

Suppose two nonnegative loads x1 and x2 have exactly known total s=3. A second channel measures a small contrast:

d=x1-x2; z=0.01*d+e; |e|<=0.02.

All values are expressed in fixed normalized units. The recorded z is 0.03. The inverse relations are x1=(3+d)/2 and x2=(3-d)/2, so nonnegativity gives -3<=d<=3. The error account gives 1<=d<=5; jointly, the compatible contrasts are 1<=d<=3.

If the question concerns the total or whether d is positive, this already answers it. A simulation that needs one nominal split must introduce a selection. Suppose its stated reconstruction requirement is to limit the contribution of channel error to at most 1 contrast unit, while accepting at most one-half shrinkage toward balanced loads. This is a modeling preference for the nominal input, not additional evidence that the loads are equal.

Use the objective

J_lambda(d)=(0.01*d-0.03)^2 + lambda*d^2; -3<=d<=3.

Its unconstrained minimizer, whenever it lies in that interval, is

d_lambda=0.01*z/(0.0001+lambda).

The data-error contribution is bounded by 0.01*0.02/(0.0001+lambda). Making it at most 1 requires lambda>=0.0001. The exact-data shrinkage fraction is lambda/(0.0001+lambda); making it at most one-half requires lambda<=0.0001. These two declared requirements select lambda=0.0001.

The nominal result is d_lambda=1.5, hence (x1,x2)=(2.25,0.75). Its predicted contrast record is 0.015, leaving residual 0.015 within the supplied error bound. For this fixed strength, the interior reconstruction gain is 50, compared with 100 for direct inversion. Projection onto the admitted interval cannot increase that gain.

The bias matters. If the underlying contrast were d=1 and the error e=0.02, direct inversion would return 3 and this regularized rule would return 1.5. If the underlying contrast were d=3 with e=0, the same observed record would make direct inversion correct and the regularized rule would understate the contrast by 1.5. These are two constructed compatible cases, not an empirical accuracy comparison.

For a worst-case guarantee from these records, retain [1,3]. Its midpoint 2 has maximum absolute error 1 over that interval, while the selected nominal value 1.5 has maximum error 1.5. The midpoint is the better choice for that different criterion. Regularization is justified here by the declared response and shrinkage requirements, not by a claim that it improves every error criterion.

Changed condition. A new acquisition reports the same z=0.03 with supported error bound 0.002. The compatible interval is now [2.8,3]. Direct inversion has data-error contribution at most 0.2, already below the allowed 1. Choose the weakest penalty meeting that requirement: lambda=0 now suffices and introduces no shrinkage. It returns d=3 with the interval [2.8,3]; a receiver minimizing worst-case absolute error could instead use 2.9.

Keeping the old strength would return d=1.5 and residual 0.015, incompatible with the new error bound. The changed observation condition, not more accurate minimization of the old objective, changes the useful formulation.

MMP.12:5.2 - A circulation selected away by minimum norm

Three stores exchange material around a directed cycle. Let the nonnegative transfers be a from the first to the second, b from the second to the third, and c from the third to the first. Their stock changes are

F(a,b,c)=(c-a, a-b, b-c).

Observed zero stock changes allow every (a,b,c)=(t,t,t) with t>=0. Minimizing a^2+b^2+c^2 selects t=0. The minimizer is unique, yet a circulation of one unit gives exactly the same records. Minimum norm has supplied a nominal no-circulation choice, not evidence that no transfer occurred.

A question about net stock change is already settled. A question about gross transported amount 3*t remains unresolved unless the subject account supplies another restriction or observation. If the stores can circulate material during unchanged stocks, using the minimum-norm answer as measured throughput would erase the quantity being sought.

MMP.12:5.3 - An accumulated quantity with an unstable derivative

In a continuous model on the normalized time interval [0,2*pi], accumulated activity is N(t), and its rate is r(t)=N’(t). Consider

N(t)=2*t; N_k(t)=2*t + sin(k*t)/k

for positive integers k. Their maximum difference is 1/k, tending to zero. Their rates are 2 and 2+cos(k*t), whose maximum difference remains 1. Both accumulated curves are nondecreasing. Thus nonnegativity of the rate does not remove this instability in the maximum norm.

A justified bound on rapid rate variation, or a penalty on changes in the rate, can suppress the oscillatory alternative. It also risks suppressing a real short surge. Specify which temporal detail the receiving use needs before selecting that structure.

If the use needs only the total over this interval, both curves give 4*pi. Recover that target from the endpoints without differentiating. For a fixed sampling interval a finite-difference inverse is continuous, but its error amplification grows as the interval shrinks; this differs from the discontinuity of the function-space inverse just exhibited.

MMP.12:6 - Bias-Annotation

The quadratic case makes sensitivity and shrinkage calculable. It does not make quadratic penalties appropriate for every unknown. A smoothness preference can remove discontinuities; a sparsity preference depends on the representation; a learned preference can lose features absent from its construction cases.

The method treats the subject grounds of additional structure as a separate contribution. Its mathematical examples establish consequences of stated premises, not the suitability of those premises for a particular physical or organizational situation. More consequential use can require stronger subject evidence or a validated target bound; ordinary use can stop at an already sufficient comparison.

MMP.12:7 - Conformance Checklist

• The target and forward relation are recoverable, including consequential unknown influences, units and record-error assumptions.
• Indistinguishable alternatives and error amplification are distinguished; a sufficient target is used without demanding full recovery.
• Each hard restriction or penalty has a subject justification or an explicit nominal-selection purpose. Its excluded or discouraged variation is identifiable.
• The strength choice exposes both sensitivity reduction and target loss. A changed observation condition is carried through that choice.
• Existence, uniqueness, stability, vanishing-noise recovery and solver convergence are claimed only at the scope established for the formulation.
• The returned target retains consequential dependence on added structure and the uncertainty needed for its receiving use.

MMP.12:8 - Common Anti-Patterns and How to Avoid Them

MMP.12:9 - Consequences

The practitioner obtains a reconstruction problem with visible reasons for its additional structure. Some questions become cheaper because a stable total or bound replaces full reconstruction. Others gain a useful nominal estimate at the cost of explicitly accepted bias.

The formulation also locates a failed use: unsupported regularity, an unresolved invisible direction, an inadequate forward relation or insufficient numerical control call for different contributions. Increasing regularization or computation indiscriminately does not resolve them.

MMP.12:10 - Architectural Rationale

The forward relation determines what the records respond to. The target determines which unresolved variations matter. Regularization determines which remaining variations the reconstruction discourages. Keeping these three contributions separate explains why one fitted answer can be useful for a nominal simulation yet inadequate as evidence about the subject.

This method extends compatible-case interpretation by constructing the recovery rule, choosing its strength and analyzing its induced loss. The distinction from computational approximation preserves the ability to solve the chosen equations accurately while still revising their recovery assumptions.

MMP.12:11 - SoTA-Echoing

How should a needed reconstruction be stabilized at the available accuracy? The selected line separates propagated error from regularization bias and chooses strength for their receiving use. The serious defaults are direct inversion or least-squares fitting, and the sufficient-bound route of C.16.IR. In :5.1, elementary calculations expose the direct gain of 100 and the selected gain of 50 at comparable effort, with shrinkage as the accepted cost. The bound wins when it settles the question; direct inversion wins after the tighter observation. Adopt this conditional choice in :4.2–4.5 rather than a default penalty. Clason, Regularization of Inverse Problems, arXiv:2001.00617v2, Chapter 4, equation (26), and Chapters 6–7 supplies the mathematical error decomposition and parameter-dependent recovery line, including iterative regularization. Its operator assumptions delimit those results; it does not justify a subject penalty. Reopen the choice when a simpler recovery or bound meets the same target conditions, or when changed error or model structure defeats the selected strength.

When can newer learned reconstruction change that choice? Adapt the distinction between reconstruction fit and recovery guarantees from Bednarski and Roith, Introduction to Regularization and Learning Methods for Inverse Problems, arXiv:2508.18178v1, §§1.3–1.4, 2.3 and 3.2–3.3. Its mathematical treatment is a best-known-line candidate for comparing classical and data-dependent regularization: training distributions and parameter rules matter to the resulting guarantee. This changes :4.3–4.5 by retaining the added structure and its recovery conditions when R is learned. A trained replacement is a serious alternative when a simple penalty loses important structure, but requires a target-specific comparison including its extra preparation cost.

Hertrich et al., Learning Regularization Functionals for Inverse Problems: A Comparative Study, arXiv:2510.01755v1, §§5.1–5.5 supplies bounded rival and failure evidence: its imaging comparisons show dependence on training, task and cost, and report attractive reconstructed geometry that differs from ground truth. Reject transferring an imaging ranking into a general reconstruction rule. Retain its action-changing lesson in :4.3: compare the actual required feature under applicable conditions. Reopen when a learned alternative preserves that feature better at justified total effort, or when a changed observation model or subject domain invalidates the comparison.

MMP.12:12 - Relations

• Uses MMP.11 to construct the model family from which the forward relation is assembled, and MMP.7 when a probability law of records is needed.
• Uses C.16.MR for a missing measurement relation and C.16.IR for compatible alternatives and sufficient target bounds. Adds regularized formulation, strength selection and analysis of induced sensitivity and loss.
• Uses MATH.20 for error and target bounds. CMP.6 and CMP.8 supply an iterative obtaining method or controlled approximate computation for the selected problem.
• Supplies statistical inference with the target, observation account and unresolved dependence on added assumptions when an estimate’s probability or uncertainty is needed. Supplies observation design with the distinctions an additional observation would need to resolve.
• Returns to B.5.RR and B.5.MPC.R when the required result changes the mathematical, subject or computational contribution. C.39 can use that result to construct a missing contribution to obtaining or using observations; C.32.MWA.PP compares its working arrangement.

MMP.12:End

MMP.13 - Infer Unknowns Statistically under a Stated Observation Model

Type: Method pattern Status: Stable Normativity: Normative within the stated use

MMP.13:1 - Problem frame

Use this pattern when a probability model describes how records arise, and the work needs an estimate, an interval or a probability about something not directly known. The wanted quantity may be a failure probability, a mean before measurement error, a response for a different mix of cases, or a future outcome.

Start by writing the sentence the result must support. “This procedure covers the fixed unknown in at least 95% of its modeled repetitions” differs from “the unknown lies here with 95% posterior probability under these assumptions.” A probability about the next observation is another question. Choose the inferential construction that supports the required sentence.

The result is an estimator or posterior and its consequence for the requested target, with an interpretable account of uncertainty. It exposes which assumptions make the conclusion possible and which change would require recalculation. Reporting a fitted coefficient and a number labeled “error” can otherwise leave the actual question unanswered.

This is statistical inference within mathematical modeling. MMP.7 supplies the law of the records; MMP.12 supplies unresolved recovery ambiguities and any regularization choices. The present method constructs what may be concluded under that law and additional inferential assumptions. Selecting an action also needs its consequences and preferences, supplied by a decision method such as MMP.8.

The reader needs conditional probability, expectations and quantiles; continuous models also use integration. A mathematically qualified collaborator can supply those operations when the practitioner can specify the observation process and interpret the target.

Use an existing sufficient inference directly. If a compatible range already settles the question, C.16.IR can end the work without a probability model. A missing observation law returns to MMP.7; constructing a numerical sampler for an already specified posterior belongs to CMP.9.

MMP.13:2 - Problem

The distribution of the observed records does not itself choose an estimator, an uncertainty statement or a prior. Those additions can produce different useful conclusions from the same data.

The inferential target may also differ from the fitted parameters. A nonlinear function of parameter estimates can give a different answer from averaging that function over a posterior. Shared uncertainty can remain after many repeated readings. A narrow interval for an expected response can coexist with a wide distribution for the next response.

The problem is to construct the inference needed by the receiving use while preserving its sampling, conditioning and approximation assumptions. Changing one of those assumptions must change the relevant calculation, rather than only the qualification attached to an unchanged number.

MMP.13:3 - Forces

MMP.13:4 - Solution

Specify the target and the intended uncertainty claim. Combine the record law with the inferential assumptions needed for that claim, derive the estimator or posterior, and propagate it to the target. Carry any actual consequential revision through the affected calculation. Compare a plausible alternative assumption only when doing so can change the use or returned claim. Return the sufficient result at the scope the calculation supports.

MMP.13:4.1 - Fix the target before choosing a fitting routine

Write q=g(theta), where theta denotes the unknowns in the observation model. It can index a distribution or unknown function, not just a finite vector. State the population, conditions and time range that make q meaningful. Distinguish unknowns needed only to explain the records from the target being returned.

For prediction, instead name the new outcome Y_new and its observing conditions. A conditional mean of Y_new is a function of theta; the realized Y_new also varies under the model. A proposed intervention requires a supplied causal identification argument before a fitted association can be interpreted as its effect.

Select the smallest result that changes the next use. It may be a point estimate with a stated error property, an upper confidence bound, a posterior probability of a threshold, or a predictive distribution. A threshold action still needs its loss or decision rule; a probability or confidence level does not choose that action by itself.

MMP.13:4.2 - Recover the law and the assumptions being added

Take the joint law P_theta of the records Y from MMP.7. Retain its inclusion, censoring, dependence and stopping conditions. Where a common probability-mass or density representation exists, inserting the observed y gives the likelihood L(theta;y)=p_theta(y).

The likelihood compares how parameter values account for the same records. It is not a probability distribution over theta merely because it can be plotted or maximized. Likewise, records in separate rows are not necessarily independent observations.

Use the existing ambiguity result from MMP.12 or C.16.IR. If two parameter values give the same observation law and different q, the records cannot distinguish that target. A prior or restriction may support a conditional answer, whose dependence on that addition remains visible.

Choose the inferential branch by the claim required. Frequentist construction assesses a data-to-answer rule across the specified observation law at fixed unknown values. Bayesian construction adds a prior law and conditions their joint model on the observed records. Neither branch removes the need for a justified observation model.

If the same data choose a model, tuning value or prior hyperparameter, include that adaptation in the inference being claimed. Treating an estimated quantity as externally known can understate uncertainty. A sensitivity comparison can instead hold the data fixed and show the consequence of several explicitly conditional assumptions.

MMP.13:4.3 - Construct an estimator and its frequentist uncertainty when needed

An estimator is a rule q_hat=T(Y), chosen before its repeated-use properties are assessed. Derive it from the target and observation law: for example, invert an observable expectation, solve an estimating equation, or maximize a likelihood and obtain the required function of the fitted values. State the selected property, such as a controlled error probability or mean squared error. Likelihood maximization alone supplies no interval.

One general confidence construction chooses, for each admitted theta, an acceptance region A_theta of possible records such that

P_theta(Y in A_theta) >= 1-alpha.

After observing y, retain those theta for which y lies in A_theta, and map them through g. The resulting set C(y) has coverage

P_theta(g(theta) in C(Y)) >= 1-alpha

under the stated law. A target-specific statistic can make this much cheaper than constructing a set for every nuisance parameter. The binomial inversion in :5.1 is a small example.

Coverage describes the procedure across repetitions allowed by the model, at fixed theta. It does not assign posterior probability to the parameter in the one observed interval. Nor is a confidence set the set of all logically possible values: a value outside it may still assign a small positive probability to these records.

Use an exact distribution or justified pivot when available. An asymptotic approximation needs its sample-size, regularity and boundary conditions. A bootstrap needs a resampling unit and mechanism that represent the dependence and the fitted procedure; resampling individual readings cannot reproduce an omitted common calibration error. Retain approximate coverage as approximate unless a stronger result is available.

The observation plan belongs to the guarantee. An interval justified at a fixed sample size need not preserve its coverage when repeatedly inspected until it looks favorable. Use a method valid for the actual stopping plan. For countably many looks, one conservative construction allocates error probabilities alpha_j with sum at most alpha to valid per-look intervals; the union bound gives simultaneous coverage. An applicable confidence-sequence method can provide a less conservative construction. This is a choice of inferential guarantee, not a requirement to collect more data.

MMP.13:4.4 - Construct a posterior when conditional probability is needed

Supply a prior probability law Pi with its subject meaning. Reweight that law by the observed likelihood and normalize, preserving any point masses and continuous parts. For a set A to which the prior assigns a probability,

Pi_y(A) = integral_A L(theta;y) Pi(dtheta) / integral L(u;y) Pi(du).

Pi_y denotes the posterior law. Integration against Pi means combining values under the actual prior: a sum for discrete unknowns, integration against a density where one exists, and both contributions for a mixture. A function-valued unknown requires a specified prior law and likelihood on that space, not an assumed ordinary density.

The denominator must be finite and positive; an unnormalized expression alone does not establish a posterior probability law. For a real vector theta whose prior has a density pi(theta) with respect to ordinary volume dtheta, the posterior density is

pi(theta | y) = L(theta;y)*pi(theta) / integral L(u;y)*pi(u) du.

This density formula is a representation of the preceding law construction under that condition. A uniform prior depends on the parameterization, and an improper prior requires a separate argument that the posterior exists.

For example, give a failure probability p prior mass 1/2 at p=0 and a uniform distribution on [0,1] for the remaining 1/2. One failure-free Bernoulli observation has likelihood 1-p. The unnormalized atom has mass 1/2; the weighted continuous part has mass 1/4. Normalizing by 3/4 leaves posterior mass 2/3 at zero. The remaining 1/3 has conditional density 2*(1-p) on [0,1]. Using only the ordinary density would lose the atom.

Retain joint dependence when removing nuisance unknowns. Sum or integrate the joint posterior over them, or calculate g(theta) from joint posterior draws. Independently combining draws from marginal distributions changes the joint law unless independence is established.

Derive the posterior of q through that transformation. For a set B,

P(q in B | y) = integral 1{g(theta) in B} Pi_y(dtheta).

A credible set has its stated posterior probability under this model and prior. A posterior mean, median or quantile is a chosen summary of that law. In general, E[g(theta)|y] differs from g(E[theta|y]); :5.1 computes the difference.

A penalized optimum from MMP.12 can coincide with a posterior mode when its objective represents the chosen likelihood and prior. That optimum still does not supply the posterior spread. If a prior resolves an otherwise unidentified difference, preserve that source of the resolution in the returned result.

MMP.13:4.5 - Propagate uncertainty to the actual receiving quantity

For a posterior predictive result, construct the law of the new observation under its specified conditions and average it over the posterior law:

P(Y_new in B | y) = integral P(Y_new in B | theta,y) Pi_y(dtheta).

In the density case of :4.4, Pi_y(dtheta) is pi(theta | y) dtheta. For a mixed posterior, include its atoms as well as its continuous contribution.

This includes both uncertainty in unknowns and the modeled variation of a new outcome. Where variances exist, the decomposition is

Var(Y_new|y) = E[Var(Y_new|theta,y)|y] + Var(E[Y_new|theta,y]|y).

The second term alone concerns uncertainty in the conditional mean. It is not the full predictive variance.

A frequentist prediction interval also needs the joint law of the original and future records. Derive a prediction error or another statistic with the needed coverage. Sharing a calibration influence with the old readings and using a fresh calibration produce different prediction problems, as in :5.2.

For a posterior of a real parameter vector theta with covariance V, the linear target q=a^T*theta has posterior variance a^T*V*a. A frequentist covariance calculation instead uses the sampling covariance of the joint estimator, retaining its repeated-use interpretation. For a nonlinear target, propagate the joint posterior or derive uncertainty from the estimator’s sampling law; an approximation needs its own conditions. A confidence set for theta can be mapped through g to obtain a confidence set for q; projecting a large joint set can be conservative. A marginal interval for each coordinate does not automatically give simultaneous coverage for a function of them.

Then obtain the numerical answer with the needed accuracy. CMP.8 supplies controlled approximate computation; CMP.9 supplies a sampler or randomized computational estimator. The posterior distribution, a confidence procedure and the algorithm approximating their consequences are different results. More posterior draws can reduce Monte Carlo error in a computed mean while leaving the posterior uncertainty about q unchanged.

MMP.13:4.6 - Test the claimed consequence and revise the assumption that matters

Check the derivation at the level needed for its use. In a finite model, normalization, enumeration or an exact calculation may suffice. For an approximate frequentist procedure, simulation at fixed parameter values can expose bias or coverage failures under the assumed law. For a posterior computation, a suitable simulation-based calibration check draws parameters and data from their joint model and tests the inference computation. These are different checks. Agreement under a model does not establish that the model describes the subject.

Select such work for an unresolved claim. A routine use of an applicable exact result need not become a new simulation study, and simulated datasets are not additional subject observations.

When a premise or target actually changes, recalculate the affected target and uncertainty; reuse unaffected results. Compare a plausible alternative dependence, prior scale, inclusion probability or target-population weight only when that comparison can change the intended use or returned claim. State whether an altered result comes from changed records, changed assumptions or a changed target, and keep a material disagreement visible. A sufficient inference under unchanged grounds needs no invented alternative.

If observed failures call for a richer model, return the model-criticism and revision question to the relevant subject method and MMP.11. If the desired effect is not causally identified, return that identification question. Designing a new observation is a further choice when the existing result is insufficient; it is not part of every inference.

MMP.13:4.7 - Return the conclusion at its established meaning

Return the target, the estimate or distribution, the meaning of its uncertainty, and the assumptions whose variation changes its use. Identify a numerical approximation limit separately. Give the receiver enough to distinguish a frequentist coverage claim, a posterior probability and a predictive claim without consulting the fitting software.

Use the sufficient result. C.11.DUA helps compare another observation, a changed formulation, more computation or action with remaining uncertainty when that choice is live. B.5.RR carries a changed premise through the reasoning; MMP.8 receives inference when an actual choice needs its consequences and preferences.

A methodological use can revise how a working method treats uncertainty: retain a shared influence, compute the requested derived quantity, or stop demanding an unidentified parameter when a sufficient consequence is available.

MMP.13:5 - Archetypal Grounding

MMP.13:5.1 - Zero recorded failures: two different uncertainty claims

A device is tested for a fixed four operations. All failures are recorded, and the supplied model gives independent Bernoulli outcomes with one unchanged failure probability p. No failures occur, so K=0 and the likelihood is proportional to (1-p)^4.

Suppose the requested target is the probability of at least one failure in two further operations under the same condition:

q=1-(1-p)^2.

The future operations are assumed independent conditional on the same p. The question concerns q, rather than every detail of an operating model.

Frequentist construction. The likelihood estimate is p_hat=K/4=0. Inserting it into the familiar normal standard-error expression gives zero estimated standard error and the interval [0,0]. At p=0.2, zero failures occur with probability 0.8^4=0.4096, and [0,0] excludes the true p on every such occurrence. This alone rules out 95% coverage.

For a one-sided 95% binomial upper confidence procedure, invert the lower-tail probability: for k<4 choose U(k) satisfying

P_(p=U(k))(K<=k)=0.05,

and set U(4)=1. Binomial test inversion gives coverage at least 95%, allowing conservatism from discreteness. At k=0,

(1-U)^4=0.05; U=1-0.05^(1/4)=0.5271.

Since q increases with p, its upper confidence limit is

1-(1-U)^2=1-sqrt(0.05)=0.7764.

This is the result of a covering procedure, not a statement that q has a 95% probability of being below 0.7764 after these records.

Bayesian construction. Choose a uniform prior for p on [0,1]. Multiplying and normalizing gives posterior density

pi(p|K=0)=5*(1-p)^4; 0<=p<=1.

Thus E[p|K=0]=1/6, and the posterior 95% upper quantile of p is 1-0.05^(1/5)=0.4507. Transforming that quantile gives a posterior 95% upper quantile of q of 1-0.05^(2/5)=0.6983. Its smaller value does not make it a uniformly better confidence limit; it expresses a different conditional claim with a prior.

To obtain the probability of a failure in the next pair, average q itself:

E[q|K=0] = 1 - integral_0^1 (1-p)^2*5*(1-p)^4 dp = 2/7.

Using the posterior mean of p first would give 1-(5/6)^2=11/36, a different value. The posterior predictive probability is 2/7; the event of a failure in that pair remains a binary future outcome.

Changed assumption. Hold the four records fixed but replace the uniform prior with density 9*(1-p)^8, favoring lower failure probabilities. The posterior becomes 13*(1-p)^12. The posterior probability of p<0.2 changes from 1-0.8^5=0.67232 to 1-0.8^13=0.94502; the predictive probability for a failure in the next pair becomes 2/15.

These changes come entirely from the prior. They neither add operations to the observed test nor establish that the new prior is appropriate. If its relevance is unresolved, return the conditional results and their difference. The frequentist bound remains available without that prior under the original fixed-sample observation model.

MMP.13:5.2 - A common calibration error survives averaging

Four readings in fixed units are 9, 10, 10 and 11, giving mean 10. Initially suppose

Y_i=mu+epsilon_i; epsilon_i independently Normal(0,1).

The noise variance 1 is supplied, not estimated from these four values. The sample mean is an estimator of mu with variance 1/4. An exact normal 95% confidence procedure uses mean(Y) +/- c/2, where c is the 0.975 standard-normal quantile, approximately 1.96. The realized interval is approximately [9.020,10.980].

A prediction interval for one independent future reading uses the error Y_new-mean(Y), whose variance is 1+1/4. Its realized 95% interval is approximately [7.809,12.191]. Uncertainty about the mean and variation of the future reading require different intervals even before any model revision.

Now revise the calibration account:

Y_i=mu+B+epsilon_i; B~Normal(0,1).

B is independent of the individual errors, shared by all readings in one setup, and drawn afresh across the repeated setups used to define the coverage claim. Then

Var(mean(Y))=1+1/4.

The confidence interval for mu widens to [7.809,12.191]. Treating B as a fresh independent error on each row would instead give variance 2/4 and understate the uncertainty. More readings in this same setup reduce the individual-noise term but leave the calibration term.

Prediction also depends on what stays shared. A new reading in the same setup has the same B, which cancels in Y_new-mean(Y); the prediction-error variance remains 1+1/4. A reading in a new setup with independent B_new has variance 1+1+1+1/4=3.25 for that error, giving approximately [6.467,13.533].

These are coverage statements over the stated repeated-observation law. If B is only an unknown fixed offset with no bound or probability law, these normal intervals for mu do not follow. MMP.12 then retains the unresolved separation of mu and B; assigning B a distribution is an additional modeling contribution.

MMP.13:5.3 - Infer the rate for the receiving workload

Two classes of requests have different probabilities of finishing by a deadline. For class A, seven of eight observed requests finish; for class B, one of eight finishes. Assume fixed sample counts, independent Bernoulli outcomes within each class, independent class data, and independent uniform priors for p_A and p_B.

The posterior densities are proportional to p_A^7*(1-p_A) and p_B*(1-p_B)^7: Beta(8,2) and Beta(2,8). Their means are 0.8 and 0.2, and each variance is 4/275. The posteriors are independent under the stated construction.

For a future workload selecting the two classes equally, the conditional completion probability is q_old=(p_A+p_B)/2, with posterior mean 0.5. Now change only the receiving workload: its known class proportions are one-quarter A and three-quarters B. The target becomes

q_new=p_A/4+3*p_B/4.

Its posterior mean is 0.35 and its posterior variance is

(1/4)^2*(4/275)+(3/4)^2*(4/275)=1/110.

No new fitting is needed for this changed target. Carrying forward 0.5 would answer for the old mixture. If the target were an individual future completion indicator, its posterior predictive probability would be 0.35 and its variance 0.35*0.65, rather than 1/110.

The transfer assumes that the within-class probabilities remain applicable. Changed operating conditions require their own relation. Uncertain class proportions or a shared influence on the two probabilities require joint uncertainty, rather than the independent weighted-variance calculation above.

MMP.13:6 - Bias-Annotation

The worked laws are simple enough for exact calculations. Real records can have selection, dependence, weak identification or model mismatch that a convenient binomial or normal family conceals. The resulting uncertainty is conditional on the assumptions represented; it does not automatically include an omitted mechanism.

Bayesian and frequentist results are compared by the conclusions they support. A narrower interval alone cannot rank different probability meanings. The choice of prior, coverage domain and receiving target is part of the inference, even when software supplies defaults.

MMP.13:7 - Conformance Checklist

• The inferential target and its subject conditions are stated separately from nuisance unknowns and the subsequent decision.
• The observation law retains consequential recording, dependence and stopping assumptions.
• The estimator’s repeated-use claim or the posterior’s conditioning assumptions are recoverable; normalization or coverage has an applicable basis.
• Joint uncertainty is propagated to the actual target. Parameter uncertainty, future-observation variation and numerical error are distinguished.
• Any actual consequential revision is propagated through the affected target and uncertainty calculation. A sensitivity comparison addresses a live alternative that can change the use or returned claim; an unchanged sufficient inference needs none.
• The returned conclusion is sufficient for its use or names the missing contribution without requiring an automatic new experiment.

MMP.13:8 - Common Anti-Patterns and How to Avoid Them

MMP.13:9 - Consequences

The receiver gets an estimate or distribution with a usable meaning for its uncertainty. A needed probability, bound or prediction can be obtained without recovering every parameter, and a changed target can sometimes reuse the existing joint inference.

The method also makes the cost of assumptions visible. Another record, another prior and another numerical approximation change different parts of the answer. A sufficient inference can be used while a stronger claim remains unresolved.

MMP.13:10 - Architectural Rationale

The observation model describes how records vary. The inferential construction determines how those records support conclusions about unknowns. Target propagation determines which consequence reaches the receiving use. Their separation permits a computation to be correct while its prior, target or observation assumption is still reconsidered.

This preserves the distinction from inverse-problem regularization, numerical computation, causal identification, model criticism and decision making. Those methods supply or receive particular contributions; none is obtained merely by fitting a statistical model.

MMP.13:11 - SoTA-Echoing

Carrying fitting into the required conclusion. Adopt target-specific propagation of joint uncertainty rather than reporting a coefficient table or substituting parameter means. Gelman, Vehtari and McElreath, Statistical Workflow, §§1.3–1.6 and 1.10, author manuscript dated 5 December 2025 for the 2026 article explains how dependencies, derived quantities and calibration affect both Bayesian and non-Bayesian workflows. This changes :4.1 and :4.5–4.6. In :5.1 and :5.3 the extra calculation is small and corrects the actual requested probability or workload target. The source does not rank one inferential philosophy above the other, nor validate a particular subject model. Reopen when another construction gives the same target meaning and uncertainty at lower effort, or when changed dependencies defeat the propagation.

Uncertainty near a parameter boundary. The serious default is the normal standard-error interval, which collapses in :5.1. Adopt binomial tail inversion when its fixed-sample coverage is the requirement; retain a posterior alternative when its prior-conditional probability is wanted. NIST/SEMATECH e-Handbook, §7.2.4.1 supplies the explicit comparator and small-sample construction. The few tail calculations cost little here and repair the degenerate result; discreteness can make coverage conservative. A posterior requires the additional prior and can change materially with it. This comparison governs :4.3–4.4 and :5.1, not a universal preference for exact intervals. Reopen for a changed sampling law or a justified approximation that meets the same coverage need more efficiently.

Inference when observations determine stopping. Adapt simultaneous coverage as a conditional alternative to fixed-time intervals in :4.3. Howard et al., Time-uniform, nonparametric, nonasymptotic confidence sequences, §1 and equation (1), arXiv:1810.08240v9 supplies the stronger guarantee and its cost in interval width. Its stated stochastic conditions still apply. A fixed-time method remains sufficient for its own observation plan; sequential machinery is unnecessary there. Reopen when the actual stopping rule changes or a narrower valid construction improves the needed inference at comparable effort.

MMP.13:12 - Relations

• Uses MMP.7 for the law of recorded outcomes and MMP.11 for the model family when that family needs construction.
• Uses MMP.12 and C.16.IR for unresolved recovery distinctions and sufficient target bounds. Adds estimator or posterior construction and the meaning of inferential uncertainty.
• Uses CMP.8 and CMP.9 to obtain numerical consequences of the selected inference, with computational error separate from inferential uncertainty.
• Supplies prediction and model criticism with conditional consequences that can be compared with relevant observations. A causal interpretation additionally needs its identification argument under C.28.
• Supplies MMP.8 with estimates or distributions used in a decision; C.11.DUA compares the value of further information or computation when needed.
• Uses B.5.RR to propagate a changed premise. C.39 can use the resulting inference to construct the changed contribution to obtaining or interpreting records; A.3.1.MV distinguishes a Method variant from an implementation correction.

MMP.13:End

MMP.14 - Find and Repair a Mathematical Model’s Failed Predictions

Type: Method pattern Status: Stable Normativity: Normative within the stated use

MMP.14:1 - Problem frame

Use this pattern when a mathematical model’s predictions leave a consequential feature of the available observations unexplained, or when a proposed use makes such a discrepancy worth examining. A model may reproduce an overall average while missing conditional responses, bursts, extremes or the records that a selection rule actually permits.

Start with one needed prediction and a feature whose failure would change its use. For overload, compare relevant tails or sequences, not only the fitted mean. Construct that feature under the model on comparable records. The first result can be a localized discrepancy, a repaired calculation, or a sufficient reason to retain a narrower use.

The principal result is a changed model, the assumption changed, and the consequence that must be recalculated. A supported restriction of use or an unresolved choice between repairs can also be the useful result. A small residual is not a certificate of adequacy; a nonzero residual is not automatically a model failure.

This is model criticism within mathematical modeling. MMP.7 supplies the recording law, MMP.11 the available model family, and MMP.13 inference within a stated model. Here the work constructs and interprets comparisons that can change that model. A physical, biological, economic or other subject method supplies the meaning and admissibility of the proposed repair. Predictive improvement alone identifies neither a causal mechanism nor an intervention effect.

Do not reopen an adequate application merely to run a standard collection of diagnostics. An established calculation, bound or checked prediction may already suffice under unchanged conditions. C.11.DUA selects further checking when its possible outcomes can change the answer, claim or warranted use. New observations, model enlargement and simulation are not routine prerequisites.

MMP.14:2 - Problem

An optimizer can improve fit without repairing the needed prediction. A model can fail a broad diagnostic while retaining a sufficient narrower answer. Apparent disagreement can also come from comparing latent quantities with selected records, numerical approximation, or ordinary model-permitted variation.

Even a real discrepancy rarely names its cause. Large residuals might reflect a missing predictor, varying noise, dependence, selection or an erroneous observation. Wholesale replacement hides the changed assumption; making every observed feature look ordinary can fit chance patterns.

The working problem is to construct a comparison that can expose the relevant mismatch, trace it to a revisable part of the model, and establish what the revision changes without treating reused data or simulated agreement as new external evidence.

MMP.14:3 - Forces

MMP.14:4 - Solution

Choose a consequential discrepancy → construct comparable predictions → locate the mismatch → change the implicated assumption → recalculate and compare the consequence → return the warranted use.

MMP.14:4.1 - Choose the prediction and the discrepancy together

State the receiving quantity and conditions: a response at specified inputs, a probability of exceeding a limit, a distribution of recorded counts, or a forecast for a given horizon and group. Name the relevant observational unit. A message, a batch of messages and a whole operating period support different comparisons.

Recover the model and its observation law. Retain units, inputs, initial conditions, exposure, recording rules and material dependence. Separate unknown parameters from assumptions such as constant response, independent errors or complete recording.

Choose a discrepancy (D) that responds to a failure relevant to this use. For example:

• Conditional bias can be exposed by mean residuals within relevant input ranges, rather than a mean over all inputs.
• A tail count (D(y)=\sum_i 1{y_i>u}) asks whether the model accounts for excursions beyond the consequential level (u).
• A run length or (D(r)=\sum_{i=2}^n r_i r_{i-1}), with residuals in their actual order, can expose dependence hidden by a histogram.
• A distribution of recorded categories can reveal that predictions concern unfiltered events while observations concern selected records.

A conditional plot, a few statistics or a direct bound can suffice. Explain which repair would change the feature. A statistic that fitting nearly forces to agree, such as the mean in a fitted constant-mean model, usually contributes little to detecting omitted structure.

The user needs the target, conditional distributions or bounds, and the comparison’s construction. Obtain a missing calculation from a mathematically qualified collaborator: for example, a record generator and a discrepancy’s reference distribution. A human or AI participant may supply it; recover the subject assumptions and interpretation before using its output.

MMP.14:4.2 - Generate the comparison the question requires

Compare quantities with the same meaning. Replicated physical states are not yet rounded, censored or selected records. Carry them through MMP.7’s recording law.

For a posterior predictive comparison, obtain joint parameter draws from MMP.13 and generate records conditionally: [ \theta^{(s)}\sim\pi_M(\theta\mid y,x),\qquad y^{\mathrm{rep},(s)}\sim p_M(y^{\mathrm{rep}}\mid x,\theta^{(s)}). ] Here (M) names the model and (x) the retained input and recording conditions. Compare (D(y)) with (D(y^{\mathrm{rep},(s)})). If the discrepancy depends on parameters, calculate (D(y,\theta^{(s)})) and (D(y^{\mathrm{rep},(s)},\theta^{(s)})) at the same draw.

Decide what repeats. New observations for existing groups with their inferred effects differ from new groups with regenerated effects. Retain or regenerate effects according to the questioned prediction. Do not independently redraw an effect that should be common to an entire batch.

A non-Bayesian comparison can use a specified parameter value, an exact conditional reference distribution that eliminates a nuisance parameter, or a fitted-model simulation. State which is used. When the reference concerns a statistic of a fitted procedure, reproduce the fitting step on each simulated dataset; a simulation that holds its fitted coefficients fixed generally answers a different question. A parametric bootstrap may approximate a reference distribution, not make its calibration exact.

For a held-out comparison, construct the forecast without using the records being predicted to fit, tune or select that forecast. Keep preprocessing inside the corresponding training operation. Choose the withheld unit and allowed information for the receiving prediction: new observations in an existing group, a whole new group, or a future block with a specified horizon. A convenient random split does not by itself establish future or new-group performance. An alternative split needs an argument that its bias and variability suffice for the intended conclusion.

Write a held-out predictive law as (Q_M(y_H\mid y_T,x)), where (T) is the available training information and (H) the withheld portion. Compare the models on the same (H), target and scoring convention. Use a joint block prediction when the question depends on within-block dependence. Pointwise scores can still answer a declared marginal prediction question; they do not test all joint behavior.

Exact enumeration or algebra may replace simulation. If a deterministic prediction and an observation each have established error bounds, compare their admissible ranges. Disjoint ranges expose an incompatibility under those bounds; overlapping ranges do not prove the model.

MMP.14:4.3 - Interpret the difference at the comparison’s actual strength

Inspect where discrepancies occur, not just whether one aggregate score changes. Compare direction, size, input region and persistence with the variations that the reference construction permits.

A posterior predictive tail fraction describes a conditional comparison under the fitted model. It is not generally a frequentist p-value with a uniform null distribution. An exact conditional tail probability, a fitted bootstrap approximation and a held-out loss have different interpretations. None is the probability that the model is false.

Account for the construction’s resolution when it can change the result. Zero exceedances in finitely many simulations does not establish a zero tail probability. Approximation error, poor sampling or an inaccurate held-out calculation can create an apparent discrepancy. CMP.8/.9 supply the relevant numerical error account. Checking recovery on data generated by the model can expose computational faults; success there does not establish the model’s correspondence with the subject.

A pattern found after searching many views remains a useful clue, but its nominal tail area is not automatically calibrated for that search. If a repeated-error guarantee matters, account for the selection or use a suitable untouched comparison. Exploratory diagnosis need not claim that guarantee.

A discrepancy can warrant restricted use, examination of one component, or rejection of a prediction. Failure to expose one means only that this check, at this resolution, has not exposed it.

MMP.14:4.4 - Change the part that explains the consequential mismatch

First trace the disputed prediction through its calculation and observation meaning. A wrong unit, event label, numerical solution or censoring convention can require correction without changing the underlying subject relation.

Then formulate a small number of plausible revisions. Show the changed mathematical component and why it can affect the discrepancy:

These are candidates, not conclusions from the symptom alone. Do not delete observations or inflate noise until everything passes. Several changes can reproduce the feature; use subject knowledge and existing discriminating observations to choose, retain conditional alternatives, or return the missing distinction.

Use MMP.11 to preserve support, constraints and known relations when extending the family. Use MMP.13 to infer the revised unknowns; MMP.12 supplies a justified restriction when added flexibility makes recovery unstable. Changed priors, constraints or noise laws can change the answer without adding information to the records.

If the remaining difference is worth a new observation, return the distinguishing prediction and feasible-design question to the applicable observation-design and subject methods. A predictive repair does not identify an intervention effect; obtain the required causal assumptions and identification separately when that is the receiving question. A narrower supported use can finish without either continuation.

MMP.14:4.5 - Recalculate the consequence and examine the repair

Recalculate the original discrepancy and the receiving quantity under the revision. Adding a term to an equation leaves both questions unfinished. Show what changes and what remains unchanged.

Compare with the previous model and a serious sufficient alternative on the same available basis. A simpler model can be preferable when its retained result suffices and the added component contributes only estimation noise or cost. A better average score can coexist with a worse consequential tail or subgroup prediction; inspect that conflict directly.

Distinguish repair construction from assessment of the repaired prediction. Reproducing the data that motivated the revision shows what the revision accommodates. An untouched set of suitable existing records can assess a forecast fixed before those records are inspected. When repeated tuning consumes that set, it becomes part of development. If the needed performance claim concerns the whole adaptive procedure, its assessment must include that adaptation, for example through a suitable outer split; it is not a test of one retrospectively selected fit.

Fresh observations are not the only useful continuation. Recalculate with available held-out records, derive the affected consequence, retain a conditional result or restrict use. C.11.DUA determines whether resolving the remaining limitation is worth its cost. Do not describe an unperformed comparison as successful.

MMP.14:4.6 - Return the model and its changed use

Return the changed relation or distribution, retained assumptions, relevant comparison, and consequence to use or recalculate. Include ambiguity where it affects use. A short explained calculation can suffice.

For a methodological use, return which discrepancy reveals the omitted distinction, how to produce comparable predictions, and which component to reconsider. For an unresolved subject use, identify the missing contribution instead of a generic demand for more data.

Recognition starts with a consequential disagreement. Assurance depends on the claim: algebra establishes a recalculated consequence; a computational check establishes its numerical execution; an appropriate comparison with observations supports the bounded subject use. Reopen when the target, regime, recording law, relevant evidence or consequential error requirement changes.

MMP.14:5 - Archetypal Grounding

The following are constructed cases. Their arithmetic demonstrates the method; the stated observations are example inputs, not reports of empirical studies.

MMP.14:5.1 - A correct overall mean conceals failed conditional predictions

Two message routes, A and B, are used under ordinary load. The target is the chance of timely delivery for each known route. In this small example every message has a complete binary record, the deadline is unchanged, and outcomes are assumed independent with a stable probability within each route and load regime.

The supplied records are partitioned before fitting. Only the fitting and diagnostic portions are opened during model construction; the assessment portion remains withheld until the revised forecasts are fixed.

The original model (M_0) ignores route: (Y_i\sim\mathrm{Bernoulli}(p)). Its maximum-likelihood estimate from the fitting portion is (\hat p=1/2). Its expected diagnostic total equals the observed total: 10 timely deliveries out of 20. That agreement does not answer the route-specific question.

Choose (D=|K_A/10-K_B/10|), where (K_A,K_B) are the timely counts in the diagnostic portion. Observed (D=0.8). Under the common-probability model, condition on the observed total (K_A+K_B=10). Then [ P(K_A=k\mid K_A+K_B=10,M_0) =\frac{\binom{10}{k}\binom{10}{10-k}}{\binom{20}{10}}. ] This reference retains the two sample sizes and removes the unknown common (p). The exact two-sided tail for (D\ge0.8) is [ \frac{2(1+100)}{184756} =\frac{101}{92378}\approx0.001093. ] It exposes a discrepancy in the common-probability account under its independence and stability assumptions. It does not identify a causal route effect. A shared disturbance confounded with route could demand a different repair.

Suppose the subject account permits route-specific response probabilities. Construct (M_1): (Y_i\mid g_i\sim\mathrm{Bernoulli}(p_{g_i})). Using the same fitting records gives (\hat p_A=0.8,\hat p_B=0.2). The changed component is the relation between the known route and the response probability, not the binary recording rule.

Recalculate the original discrepancy under the fitted (M_1), retaining the same total of 10. Conditional replicate counts have weights [ w_k=\binom{10}{k}^{{2}16^k,\qquad P(K_A=k\mid K_A+K_B=10,\hat M_1)=w_k/\sum_{j=0}}{10}w_j, ] where (16=(0.8/0.2)/(0.2/0.8)) is the fitted odds ratio. Summing (k=0,1,9,10) gives (P(D\ge0.8)=0.37348). The revised point model accommodates the diagnostic contrast. This calculation is conditional on its fitted probabilities; it neither calibrates a test of the estimated family nor independently confirms the repair.

Fix these point-probability forecasts and open the assessment portion. The sum of log probabilities of its 20 individual outcomes, using natural logarithms, is [ L_0=20\log(0.5)=-13.86294,\qquad L_1=16\log(0.8)+4\log(0.2)=-10.00805. ] Thus (M_1)’s fixed forecasts gain (3.85490) on this portion. This is an observed paired comparison, not a guaranteed future gain or a parameter-uncertainty interval. The diagnostic portion was used to propose the repair; it was not counted as untouched assessment.

For three future independent A messages under the same regime, the point forecast of at least one late delivery changes from (1-0.5^3=0.875) to (1-0.8^3=0.488). That receiving calculation must change. If uncertainty about the probabilities matters, MMP.13 must propagate it; the point calculation does not already do so.

Changed condition. The supplied operating condition now specifies high load. Before any refitting, a supplied high-load batch has 5/10 timely outcomes on each route. The ordinary-load forecasts give [ L_1^{H}=10\log(0.8)+10\log(0.2)=-18.32581, ] whereas the common (0.5) forecast still gives (-13.86294). Carrying over the repaired forecast loses (4.46287) on this batch. The earlier assessment concerned ordinary load and does not establish transfer. Keep that use boundary and examine invariance if a high-load forecast is needed. A high-load common-rate fit of (0.5) is a possible new model; its fit here is not an untouched assessment. Its three-message late-delivery forecast would again be (0.875), conditional on that rate and independence.

For the narrower question of expected timely deliveries with equal numbers of A and B under ordinary load, both fitted models give one half of the total. If only that expectation is needed and its basis suffices, this case does not require adopting the richer model or obtaining new observations. It does not make the two models’ conditional or joint predictions equivalent.

MMP.14:5.2 - Repair the law of the exported records

A candidate latency model assigns (X) uniformly to the integer values 1 through 6. An export contains only values 1, 2 and 3. Comparing these records with unconditional replications of (X) suggests too few large values.

The documented export rule, supplied independently of that discrepancy, retains a record only when (X\le3). The failed comparison omitted selection. Compose the candidate event law with the actual rule: [ P(R=j\mid\mathrm{retained}) =\frac{P(X=j)}{P(X\le3)}=\frac13,\quad j=1,2,3. ] The predicted mean of exported values is 2, not 3.5. Their probability of exceeding 2 is (1/3), not (2/3). Construct comparable replications by applying the same filter, or draw directly from this conditional law. Do not reduce the latent model’s tail merely to make the unfiltered comparison agree.

If the documented export cutoff changes to 4, the corresponding mean becomes 2.5 and the probability of an exported value exceeding 2 becomes (1/2), with no alteration to the candidate uniform latent law. These are recalculated consequences of the changed recording rule, not new empirical confirmation of that law.

Agreement within the retained range does not establish the distribution among unrecorded values. If the target is only the distribution of exported records, that unresolved tail may be irrelevant. If the target needs latent large-latency probabilities, return the missing information or assumption through MMP.12/C.16.IR; the selection correction has not recovered it.

MMP.14:5.3 - A failed numerical prediction need not require a new subject model

A normalized quantity is modeled by (u’(t)=-u(t)), (u(0)=1). A record at (t=1) is (0.370), with an established absolute recording error at most (0.005). A forward-Euler computation with step (h=1) predicts 0.

Before replacing the decay law, compare the numerical result with the model’s exact consequence (u(1)=e^{-1}\approx0.367879). Euler steps (h=1/2) and (h=1/4) give (0.25) and (0.316406); the sequence of approximations exposes a material computational error. The exact value lies in the observed admissible interval ([0.365,0.375]). Here the repair belongs to CMP.8, while this observation supplies no reason to change the decay relation.

That interval overlap establishes compatibility for this record under the error bound, not the correctness of the decay model at every time. If a relevant observation instead excludes the accurately computed consequence, the subject relation or observation account becomes live again. Numerically precise evaluation and adequate subject prediction remain separate achievements.

MMP.14:6 - Bias-Annotation

Familiar diagnostics favor failures that are easy to display. Aggregate scores can hide sparse but consequential regimes; selecting the most striking plot can exaggerate ordinary variation. A preferred causal story can make one repair appear uniquely compelled when several produce the same records.

Preserve the intended use, the role of each data portion and the compatible alternatives. Do not transfer support to unrepresented conditions.

MMP.14:7 - Conformance Checklist

• The questioned prediction and a discrepancy that can change its use are recoverable.
• Observed and predicted quantities share the relevant inputs, recording meaning, unit and dependence conditions.
• The replication, conditioning or withholding construction supports the interpretation actually claimed.
• Computational error and ordinary model-permitted variation have not been silently treated as subject-model failure.
• A proposed repair names the changed mathematical component and its subject basis; unresolved alternatives remain visible where consequential.
• The receiving consequence has been recalculated, and the role of data reused during repair is stated where it limits assessment.
• The result gives a warranted use, restriction or missing contribution, without imposing new evidence acquisition on a sufficient existing answer.

MMP.14:8 - Common Anti-Patterns and How to Avoid Them

MMP.14:9 - Consequences

The practitioner can replace “the model fits badly” with a calculable failed prediction and an explicit revised assumption. Downstream users can see which consequence changes, which remains sufficient, and which use is unsupported.

Comparable predictions cost work. Adaptive repair consumes assessment information, and extra parameters can weaken estimation. Improving one feature can leave another failure unresolved. Retain the use boundary without demanding exhaustive criticism of every model.

MMP.14:10 - Architectural Rationale

B.5.TC aligns competing accounts; B.5.RR carries a changed premise into its consequences. Neither alone constructs a conditional discrepancy distribution, a predictive replication or a withholding scheme. Those mathematical operations provide the contribution here.

MMP.13 determines inference under a model. This method examines where that model fails a needed prediction and changes a component before inference is repeated. MMP.7 prevents a change in recording from being mistaken for a change in the subject. MMP.11 supplies admissible replacement relations, while the subject method determines what those relations purport to represent.

The cases separate a missing conditional relation, a missing selection rule and an inaccurate computation.

MMP.14:11 - SoTA-Echoing

Working question. How can a practitioner find the model component that spoils a needed prediction and repair it without confusing better fit, statistical surprise and external confirmation?

Selected line. Combine discrepancy-directed predictive criticism with evaluation appropriate to the intended prediction. Use an exact conditional comparison, fitted simulation, posterior replication or held-out prediction according to the claim. No inferential framework wins independently of the question.

• Gelman, Vehtari and McElreath, Statistical Workflow, §§1.7–1.8 and 1.10–1.11. The text used is the author manuscript dated 5 December 2025 for the 2026 article, not the publisher’s typeset version. Its operative contributions here are localizing misfit through predictive comparisons, understanding changes through related models, and separating computational calibration from subject adequacy. Adopt those in :4.3–:4.5. Adapt the wider workflow to a consequential local question; fitting increasingly flexible models until no anomaly remains can absorb legitimate rare patterns.
• Stan User’s Guide 2.39, “Posterior and Prior Predictive Checks”, posterior checks, discrepancy statistics and mixed hierarchical replication. Adopt the explicit replicated-data construction and the choice of what is regenerated in :4.2. Retain the limitation that a posterior predictive tail fraction is not generally a classically calibrated p-value. A statistic largely determined by fitting can miss the omitted structure. The Bayesian construction is one available comparison, not a requirement to replace an exact conditional or sufficient deterministic argument.
• Aki Vehtari, Cross-validation FAQ, online version consulted 16 September 2026, §§3, 5 and 7–11. Adopt its separation of the prediction task, partition and loss in :4.2/:4.5, and its treatment of selection-induced bias. The closest task-matching split is a useful starting point, not an unconditional optimum: alternative partitions can trade bias for variance. Held-out performance can compare predictions without identifying the component that needs revision.

Serious alternatives on the same question. In :5.1, optimizing and checking the pooled fit costs less and answers the expected-total question. It fails to expose the conditional discrepancy relevant to forecasting an A message. A held-out log-score comparison of the two fixed forecasts supplies useful performance evidence on the supplied assessment portion, but the score alone does not explain what relation to change. The chosen construction adds the route contrast and a conditional reference, then compares the explicit repair on that same predictive question. It costs an additional fitted probability and leaves uncertainty and regime transfer unresolved; it is not superior for a target already supplied by the pooled expectation.

Reopen the choice when the target becomes a new group, a different horizon, a tail or an intervention; when dependence or selection changes; or when the error requirement makes an approximate comparison insufficient. A more elaborate model or checking scheme earns its place through the new question, not through its recency.

MMP.14:12 - Relations

• B.5.TC / B.5.RR: align accounts and revise dependent reasoning; this method supplies the model-criticism constructions.
• MMP.7: constructs the law of recorded data used for comparable predictions and repairs to selection or measurement assumptions.
• MMP.11 / MMP.10: supply admissible model families and constraint formulations when the failed prediction requires a changed relation.
• MMP.13 / MMP.12 / C.16.IR: supply inference, regularized recovery and the limits of what records resolve; a repaired fit does not remove those limits.
• CMP.8 / CMP.9 / MATH.20: obtain predictions and discrepancy calculations with the relevant numerical error or bound.
• Causal-identification and observation-design methods: receive the unresolved causal effect or distinguishing-observation question when it matters. Obtain these contributions from a suitable subject source or collaborator.
• C.11.DUA and the subject method: select worthwhile checking and establish the subject meaning of a repair. The receiving practice uses the revised prediction or restriction.

MMP.14:End

MMP.15 - Identify an Intervention Effect from Available Data

Type: Method pattern Status: Stable Normativity: Normative within the stated use

MMP.15:1 - Problem frame

Use this when the work asks what changing an action, operating condition or working method would do, and available observations or experiments might already determine the answer. A fitted association, even a precise one, does not by itself say what would happen under that change. Start by writing the intervention and the quantity needed, then ask which available probability laws and causal assumptions can determine it.

The practical gain is an expression that can be estimated from the available records, a useful bound, or an explanation of the remaining ambiguity. Identification can spare an unnecessary experiment or prevent an unsupported intervention claim.

This is a method of mathematical modeling within causal reasoning. It constructs the connection from available laws to an intervention target. C.28.MR supplies the meaning and consequences of replacing a mechanism in a given causal model; here the model’s relevant mechanisms may remain unknown. Subject methods must justify what the intervention changes, which influences may be shared, and which relations remain applicable. The method applies to physical, biological and computational systems and to human or AI working methods.

Preparation requires conditional probability, expectation and the ability to follow a mathematical argument about assumed causal relations. For graphical derivations, the practitioner or an available specialist must be able to check which paths remain open under conditioning and intervention. A specialist contribution must return the target, assumptions, required observable quantities and derivation or obstruction; a software answer alone is insufficient.

Do not reconstruct identification when an applicable result already answers the unchanged question. Use MMP.13 for estimation under an already identified expression, and C.28.MR for a consequence inside a fully supplied causal model. Design another experiment only when the existing result is insufficient and further evidence is worth obtaining. For a question about the same observed case under another action, or a joint or nested comparison of its responses, MMP.19 constructs the common-case relation and determines what the available information fixes.

MMP.15:2 - Problem

Several causal models can produce the same observable distribution and different intervention consequences. Selection adds another limitation: an experiment can identify an effect among its participants without identifying the effect in the receiving population.

The task is to determine whether the requested causal quantity follows from the available laws and stated assumptions. Failure to find an adjustment set is not failure of identification. Fitting, choosing a prior or simulating one compatible mechanism does not prove that all compatible models give the same answer.

MMP.15:3 - Forces

• The requested effect versus an unnecessarily complete model. An average contrast may be determined while individual responses or the full intervention distribution remain unknown.
• Usable assumptions versus attractive fit. Temporal ordering, excluded pathways and invariance across populations carry causal meaning that goodness of fit cannot supply.
• Available information versus convenient information. Separate experimental, selected and population laws cannot be treated as one complete joint distribution without a derivation.
• Simple sufficient construction versus broader search. Adjustment is often enough. Mediated identification or a general identification algorithm can succeed where adjustment does not, at the cost of additional structural reasoning.
• A conditional answer versus another investigation. An explicit bound or assumption-dependent result may already suffice. New observations must answer an unresolved consequential question.

MMP.15:4 - Solution

Define the target, recover the available laws and describe the admitted causal models. Derive a common expression for their intervention consequence, or demonstrate how they disagree. Limit the returned conclusion to what that argument establishes.

MMP.15:4.1 - Specify the causal quantity and what counts as the same intervention

For an action level (a), let (Y(a)) denote the outcome under the specified intervention setting the action to (a). For a target population (T), a common request is

[ \mu_a=E_T[Y(a)], \qquad \Delta=\mu_1-\mu_0. ]

Define the action, comparator, outcome, population and time horizon far enough to distinguish the actual question. A contrast of average outcomes differs from the chance that a particular unit would benefit. Identifying both marginal laws of (Y(1)) and (Y(0)) does not ordinarily identify their joint law or the distribution of individual differences.

Establish consistency: for a unit actually receiving the specified version of action (a), its observed outcome agrees with (Y(a)). Different doses, implementations or accompanying actions may give the same label different meanings. If one unit’s outcome depends on other units’ actions, represent the relevant joint assignment or policy; the notation (Y(a)) must not silently discard that dependence.

For static interventions, (P(Y\mid do(A=a))) denotes the corresponding intervention law. C.28.MR defines the change of mechanism that gives this notation its meaning. Identification below seeks that law or its required function without assuming all the changed model’s numerical mechanisms are already known.

MMP.15:4.2 - Recover the available laws and the admissible causal models

Use MMP.7 to recover how records arise. With inclusion indicator (S), selected records supply a law such as (P(A,Y,L\mid S=1)), not automatically (P(A,Y,L)). Keep assignment, actual action, measurement and inclusion distinct when those distinctions affect the target. State which variables were jointly observed, which interventions were performed, and which population each source concerns.

For identification, provisionally treat these population laws as known. This asks what unlimited data of those kinds could determine. MMP.13 handles finite-sample estimation.

Describe the admitted causal relations. An acyclic causal graph is a useful representation: directed arrows allow direct causal influence, and a shared unobserved cause can be represented explicitly or by a bidirected edge. The absence of an arrow excludes a possible influence relative to the represented variables. A good observational fit does not justify that exclusion. Time-indexed variables can express feedback across time; a theorem for acyclic graphs must not be applied unchanged to an equilibrium model with unresolved cycles.

When these causal relations or material rival mechanisms still need construction, C.28.CM develops the model family, its subject meanings and the consequences that could distinguish its members. Bring that result back to the identification question. A sufficient already supplied family needs no additional construction.

The operative identification test is this: whenever two admitted causal models induce the same available laws, must they give the same requested quantity? If the assumptions themselves conflict with the available laws, return that conflict rather than declaring a result identified through an empty model class.

MMP.15:4.3 - Derive an expression, using the simplest sufficient route

Begin with a relation that replaces an unobserved intervention quantity by available quantities under an explicit assumption. Then combine it with ordinary conditioning, multiplication and averaging. Every final factor must come from an available law or an earlier justified step. Displayed sums concern discrete variables; for continuous or mixed variables, average against their actual probability laws.

For example, let (L) be measured before action. For a mean effect in the same population as the data, suppose that within each relevant (L=l), action assignment does not select a different mean potential outcome:

[ E[Y(a)\mid L=l]=E[Y(a)\mid A=a,L=l]. ]

This mean exchangeability assumption, consistency and positive probability of each required action in the relevant strata give

[ \mu_a =\sum_l E[Y(a)\mid L=l]P(L=l) =\sum_l E[Y\mid A=a,L=l]P(L=l). ]

The first equality averages over the target strata. The second uses exchangeability to compare the same potential outcome, then consistency to replace it by an observed outcome. Average both actions with the same target weights. The different distributions (P(L\mid A=a)) generally answer an association question instead.

A common graphical justification is the back-door criterion: choose pretreatment variables that block every path into (A) capable of linking it to (Y) through other causes. In checking a path, conditioning on an intermediate non-collider blocks it; a collider, where two arrowheads meet, blocks it unless that collider or a descendant is conditioned on. Thus adding every available covariate can open a path rather than remove bias. This criterion is sufficient, not the only way to justify adjustment or identify an effect.

When adjustment of (A)’s effect is unavailable, derive other observable intermediate quantities. In a simple front-door construction, a measured mediator (M) carries every directed path from (A) to (Y); (A) to (M) has no open back-door path; and conditioning on (A) blocks all back-door paths from (M) to (Y). Under these conditions and the required support,

[ P(Y=y\mid do(A=a)) =\sum_m P(M=m\mid A=a) \sum_{a’} P(Y=y\mid M=m,A=a’)P(A=a’). ]

The outer factor identifies the effect of action on the mediator. The inner adjustment identifies the outcome law under intervention on the mediator. The pathway restrictions license combining them for the action’s total effect. Merely including a post-action measurement in a regression does not perform this construction. These classical conditions are sufficient; their failure does not establish that this functional or another identifying expression is impossible. Front-door criteria and their extension, §§2.1.5–3.

For a more involved graph or several input laws, derive a sequence of intermediate distributions using the rules of do-calculus and probability. Each exchange between observation and intervention needs the corresponding separation condition in the modified graph. A suitable identification implementation can carry out that search: provide the graph, the target and the actual input laws, then recover the returned derivation and check its required factors. The historical ID algorithm covers a specified acyclic model class with latent common causes and an observed joint law; generalized search can use several incomplete or experimental laws. Do not replace those inputs by a joint law that the records never supplied. ID algorithm, Figure 3; generalized search, §§2–3.

MMP.15:4.3.1 - Use a binary instrument for a specified local effect

Use this branch when a binary assignment or encouragement (Z) changes actual action (A\in{0,1}), and an instrumental-variable argument might identify an effect of (A). The effect of offering an action and the effect of performing it are different targets. If the receiving question requires the population average effect, determine whether the local quantity below answers it before using the result.

Let (A(z)) be the action a unit would take under assignment (z). Start with (Y(z,a)) for its outcome under assignment (z) and action (a). Use the simple argument only under these assumptions:

• Consistency and stable versions: observed (A=A(Z)) and (Y=Y(Z,A)); the specified interventions have fixed meanings and no interference between units.
• Assignment independence and support: (Z) is independent of the potential actions and outcomes, and both assignment levels have positive probability in the population supplying the law.
• Exclusion: (Y(z,a)=Y(a)); assignment has no effect on the outcome except through actual action.
• Monotonicity: (A(1)\geq A(0)) for each unit. Assignment 1 never discourages a unit that would act under assignment 0.
• Relevance and finite means: (\delta_A=E[A\mid Z=1]-E[A\mid Z=0]>0), and the required outcome expectations exist.

These premises can be more demanding than random assignment alone. Randomizing an offer can support assignment independence without establishing exclusion or monotonicity. This is the binary local-effect argument of Angrist, Imbens and Rubin, 1996, §§2–4.

The response pair ((A(0),A(1))) separates always-takers ((1,1)), never-takers ((0,0)), compliers ((0,1)) and defiers ((1,0)). These are potential-response groups; observing one assignment and action generally does not label the individual’s group. Monotonicity excludes defiers.

To derive the result, use exclusion and the binary-action identity (Y(a)=Y(0)+a{Y(1)-Y(0)}). Independence and consistency give

[ \delta_Y=E[Y\mid Z=1]-E[Y\mid Z=0] =E[(A(1)-A(0))(Y(1)-Y(0))]. ]

The same argument gives (\delta_A=E[A(1)-A(0)]). Under monotonicity, that difference is one for compliers and zero for the other admitted groups. Consequently,

[ \frac{\delta_Y}{\delta_A} =E[Y(1)-Y(0)\mid A(1)>A(0)]. ]

This Wald ratio identifies the local average treatment effect for compliers under the named instrument and population. It is neither automatically the effect for everyone who took the action nor the population average effect. Changing the encouragement can change the complier group.

If (\delta_A=0), the ratio is undefined. A small nonzero population first stage can still identify the local quantity under the assumptions, while finite-sample uncertainty and sensitivity to premise violations can be large; MMP.13 must supply suitable inference for the receiving use. If defiers are possible, the numerator and denominator instead mix oppositely signed response-group contributions. A direct (Z)-to-(Y) route, assignment dependence or interference also requires a different argument. Conditional instrument validity, continuous action and transport to another population need their own derivations; the simple unadjusted ratio does not supply them.

MMP.15:4.4 - Carry selection, transport and support through the expression

Identify which population supplies every average. For example, an experiment in (S=1) may supply action-specific outcome means by pretreatment (L), while an existing inventory supplies the target distribution (P_T(L)). If the intervention has the same meaning, assignment in the experiment is exchangeable, and the conditional potential-outcome means agree between the experimental and target populations, then

[ \mu_a=\sum_l E[Y\mid A=a,L=l,S=1]P_T(L=l). ]

This is a transport assumption followed by trial identification and target averaging. It requires trial coverage and both action levels wherever the target gives relevant weight. Equal variable names, similar marginal distributions, or randomization within the experiment do not establish the transport assumption. Selection after action or outcome requires its own recording and causal argument; it cannot automatically use this pretreatment bridge. Generalizing trial results, §§3–4.

Inspect the support needed by the chosen expression. Positivity means that a required conditional law is defined on the relevant target support; in the discrete adjustment example, (P(A=a\mid L=l)>0) whenever the target weights that stratum. For continuous variables, use the corresponding support condition. A fixed intervention value in a continuous action space may also need a continuity or other structural restriction: the observed law determines conditional responses only almost everywhere. An empty cell in a small sample does not prove a population probability is zero. A known structural exclusion is different: the missing conditional response cannot be learned there from that source.

Failure of support invalidates that expression at the excluded values. Another source, another identifying argument or an explicit structural restriction may still answer the question. Extrapolation through a response model can produce an assumption-dependent answer, but fitting or regularizing that model does not create observations at the missing support.

MMP.15:4.5 - Distinguish a proof of ambiguity from an unfinished search

To demonstrate nonidentifiability, construct two causal models satisfying the stated assumptions, reproducing all the available laws, and giving different values of the requested target. C.28.MR can calculate each model’s intervention consequence. Show both the observational agreement and the difference after intervention, as in :5.2.

A complete identification algorithm can also return an obstruction; check its class and query. A time limit or exhausted search without an applicable completeness result leaves the derivation unresolved. Failure to identify a full intervention distribution also does not by itself prove failure to identify a particular mean or contrast. ID completeness; scope of search completeness, §3.4.

When a point is not identified, use the assumptions to bound the requested quantity. Optimize it over compatible models, or derive an inequality valid for all of them. One compatible example establishes possibility; it does not establish a bound. If a bound is claimed sharp, show that compatible models attain or approach its endpoints. A bound that already settles the receiving question needs no complete reconstruction of unknown mechanisms.

A prior or added restriction may select among compatible answers; retain that dependence. It is not new evidence.

MMP.15:4.6 - Return the sufficient result and propagate consequential changes

Return the target and the assumptions under which its identifying expression, bound or ambiguity result holds. Make the data factors recoverable for estimation and uncertainty under MMP.13. Numerical evaluation belongs to an appropriate computational method; a completed optimization or simulation is not an identification proof.

Check the derivation as needed: verify a finite case, inspect a disputed graph condition, or calculate a countermodel’s observational and intervention laws. These checks establish the mathematical consequence of assumptions. Assurance that the assumptions describe the subject requires relevant subject evidence; successful fitting or self-consistent simulation does not provide it by itself.

When an actual change admits a direct pathway, changes an intervention’s version, alters selection, or changes the target population, propagate it through the affected derivation and calculation. Compare an alternative assumption when that comparison can change the intended use. Do not invent a sensitivity exercise for an already sufficient result under unchanged conditions.

C.11.DUA selects whether remaining uncertainty warrants more evidence, an explicitly conditional answer, a narrower agreed question or no further work. MMP.16 can design an observation if that further work is chosen. A mathematical identification result does not authorize or physically carry out the intervention.

MMP.15:5 - Archetypal Grounding

These constructed population probabilities make the calculations inspectable; they are not empirical results. Each case assumes fixed intervention meanings and no interference between its units. Finite records add estimation uncertainty to the assumption dependence shown here.

MMP.15:5.1 - Separate a production action from the batches receiving it

A production team asks whether enabling a stabilization mode (A=1), rather than (A=0), increases the probability of a conforming item (Y=1). Incoming batch condition (L) is recorded before mode selection. Half the target batches have (L=0), half (L=1). Mode 1 is used on 80% of (L=0) batches and 20% of (L=1) batches.

The subject account asserts consistency and mean exchangeability given (L); both modes occur in both strata. Adjustment therefore gives (\mu_0=0.5(0.10)+0.5(0.70)=0.40), (\mu_1=0.5(0.20)+0.5(0.80)=0.50), and (\Delta=0.10).

The selected groups instead give (E[Y\mid A=1]=0.8(0.20)+0.2(0.80)=0.32) and (E[Y\mid A=0]=0.2(0.10)+0.8(0.70)=0.58): an association of (-0.26). Its sign differs because the two modes receive different batch mixtures.

This positive average effect is conditional on the causal account; the table does not prove exchangeability. An already supported account and sufficient estimate need no new experiment.

MMP.15:5.2 - Use a mediator, then withdraw a pathway exclusion

A service’s command (A) can activate a retry mechanism (M), which affects successful completion (Y). Unrecorded load can affect both command selection and completion. The initial causal account permits (A\to M\to Y) and an unobserved common cause of (A,Y), but no other arrows: in particular, no direct effect of (A) on (Y) and no hidden common cause of (A,M) or (M,Y).

All three variables are binary. The available joint law has (P(A=1)=0.5), (P(M=1\mid A=0)=0.25), (P(M=1\mid A=1)=0.75), and:

There is no observed pretreatment adjustment variable for the shared load. The front-door conditions nevertheless hold. First adjust the mediator’s effect over (A): the inner means are (h(0)=0.5(0.10)+0.5(0.30)=0.20) and (h(1)=0.5(0.70)+0.5(0.90)=0.80). Then average over the mediator law induced by each command:

[ \mu_0=0.75(0.20)+0.25(0.80)=0.35,\qquad \mu_1=0.25(0.20)+0.75(0.80)=0.65. ]

Thus (\Delta=0.30). Directly comparing observed commands instead gives (0.75-0.25=0.50).

Changed condition. Inspection of the service reveals that the command can also change completion through a path bypassing retry. The revised account permits a direct (A\to Y) arrow. Retaining the previous formula is no longer justified.

The same full observational law now admits at least two answers. To demonstrate this, let unobserved (U) be a fair binary variable, let observational command selection be (A=U), and generate (M) with probability (0.25+0.50A) using independent random variation. In two alternative models, generate completion with respective probabilities

[ \text{Model I: }0.10+0.60M+0.20U,\qquad \text{Model II: }0.10+0.60M+0.20A, ]

using a further independent random draw. Every probability lies between zero and one. Because (A=U) observationally, both models give every cell of the supplied joint law.

Under (do(A=a)), (U) remains fair. Model I gives (\mu_0=0.35,\mu_1=0.65); Model II gives (\mu_0=0.25,\mu_1=0.75). Their effects are 0.30 and 0.50. Both satisfy the revised account, which permits the listed influences without requiring every permitted influence to be nonzero. Model II was excluded by the original no-direct-effect premise.

This is a proof that the revised assumptions and available law do not identify the average effect. These are ambiguity witnesses, not discovered service mechanisms. More precise estimation of the same law cannot distinguish them. A substantive restriction might; further evidence is chosen by its value for the use.

MMP.15:5.3 - Transfer an experiment to another mixture of sites

An ecological model asks for the effect of a specified irrigation change (A) on establishment of a seedling (Y), averaged across a target collection of sites. Soil stratum (L) is known from an existing inventory. A selected experiment randomized both irrigation levels within each stratum, recorded every outcome, and used the same irrigation implementations as the target question.

Suppose the subject account supports equality of conditional potential-outcome means between experimental and target sites. Experimental sites are 80% (L=0) and 20% (L=1); the target inventory is 25% (L=0) and 75% (L=1).

The experimental average effect is (0.8(0.30)+0.2(0.10)=0.26). The target means are (\mu_0=0.25(0.20)+0.75(0.60)=0.50) and (\mu_1=0.25(0.50)+0.75(0.70)=0.65). The requested effect is 0.15. Randomization identifies the comparisons inside the experiment; the transport assumption and target inventory justify the different outer average.

Now suppose the source is found to contain experimental outcomes only for (L=0); the supplied (L=1) values were extrapolations, with no justified response relation supporting them. Retain the target inventory and transport within the covered stratum. With no outcome restriction for (L=1) beyond binary outcomes, its mean effect (\delta_1) lies in ([-1,1]). Therefore

[ \Delta=0.25(0.30)+0.75\delta_1\in[-0.675,0.825]. ]

The endpoints are attainable by making every uncovered site respectively harmed or helped: ((Y(0),Y(1))=(1,0)) or ((0,1)). Those choices do not alter any available experimental outcome. The bound is sharp under these assumptions, and the sign of the target effect is not identified. The effect 0.30 remains available for the covered stratum if that is the agreed receiving question; it must not silently replace the original population target.

MMP.15:5.4 - Distinguish an offer effect, a local use effect and an overall effect

A team considers offering help with a work procedure. Let (Z=1) mean an independently randomized offer, (A=1) actual use, and (Y=1) successful completion. Assume fixed versions, no interference, exclusion, monotonicity and complete outcome recording. The following constructed population supplies one possible basis:

Randomization is independent of response group and potential outcomes. The action rates are 0.2 without the offer and 0.6 with it. The outcome means are

[ E[Y\mid Z=0]=0.2(0.60)+0.4(0.40)+0.4(0.30)=0.40, ] [ E[Y\mid Z=1]=0.2(0.60)+0.4(0.65)+0.4(0.30)=0.50. ]

The offer effect is 0.10. The first stage is 0.40, so the ratio gives (0.10/0.40=0.25), the complier mean effect. Under the stipulated full table, the population effect is instead (0.2(0.50)+0.4(0.25)+0.4(0)=0.20).

The observational law does not reveal the full table. Change the never-takers’ mean (Y(1)) from 0.30 to 0.80 while retaining all other quantities. Their action remains zero under both assignments, so this change leaves the full observed law of (Z,A,Y) unchanged and preserves the assumptions. The overall effect becomes 0.40; the identified complier effect remains 0.25. This pair of models shows why the available law does not identify the overall effect under these premises.

Now change the offer itself: it teaches a technique that can improve success without use of the help. Exclusion no longer holds. Retain the randomized offer-effect question, but withdraw the former interpretation of the ratio as the complier use effect until a revised model supplies a valid argument. Observing the same four means would not restore the missing exclusion premise.

MMP.15:6 - Bias-Annotation

Predictive success invites reading inputs as controls. Readily measured variables invite unjustified adjustment, while unmeasured common causes disappear for lack of a data column. Construct the causal account from the subject process.

Failure of a familiar criterion can instead encourage unnecessary data collection. Search more broadly when the answer matters, preserving unresolved search as unresolved. Check that calculation has not substituted a convenient population or intervention version for the requested one.

MMP.15:7 - Conformance Checklist

• The intervention, comparator and receiving quantity are defined, with consistency and interference handled where relevant.
• A local instrumental-variable effect retains its assignment, response group and assumptions; a different requested population effect remains a different target.
• The derivation uses the laws actually supplied by the recording and selection procedures.
• Every causal substitution has an assumption or applicable graphical argument; every final data factor is available on its required support.
• A nonidentifiability claim has a target-matched witness or applicable complete-method obstruction. An unfinished search is reported separately.
• A returned bound is justified for all compatible cases; sharpness is claimed only with an attainability argument.
• Estimation uncertainty, causal assumption dependence and computational approximation retain their different meanings.
• Actual consequential changes are propagated. A sufficient existing result does not trigger an obligatory new study.

These conditions recognize a constructed identification result. They do not establish the subject truth of its causal assumptions or authorize an intervention.

MMP.15:8 - Common Anti-Patterns and How to Avoid Them

MMP.15:9 - Consequences

Estimation concentrates on the required observable quantities. Nonidentifiability becomes an explicit limitation of assumptions and information, not an unexplained numerical failure.

The causal argument may demand subject knowledge unavailable in the dataset. More elaborate identification does not strengthen an unjustified premise. A conditional answer or bound can be useful without recovering every mechanism.

In methodology, this changes how a proposed improvement to a working method is assessed. Define the method change and relevant outcome, recover how cases and results were recorded, and distinguish its effect from differences in which cases received it. The same reasoning can govern a human procedure, an automated procedure or their combination, without making a population experiment mandatory for every use of the method.

MMP.15:10 - Architectural Rationale

Mechanism replacement gives intervention semantics within a model. Identification adds a different construction: showing that the needed intervention consequence is common to the causal models compatible with available laws and assumptions, or exposing where it is not.

The target-first construction avoids demanding a complete causal model when a mean, contrast or bound suffices. Separating derivation from estimation lets one identifying expression support different appropriate inferential methods. Separating source laws prevents mathematical convenience from turning selected records into information about an unobserved population.

MMP.15:11 - SoTA-Echoing

The governing question is whether the available laws and defensible causal assumptions determine the requested intervention quantity. A serious default is justified covariate adjustment; another is to fit a response model and predict after changing its action input. Adjustment is retained when it supplies a valid, economical derivation. Prediction under a fitted response model is accepted as an intervention answer only when its structural assumptions identify that meaning.

Historical completeness anchor. Shpitser and Pearl, Identification of Joint Interventional Distributions in Recursive Semi-Markovian Causal Models (AAAI 2006, UCLA report R-327), Figure 3 and the soundness/completeness argument, supply an algorithm that can return an expression or a graphical obstruction in its specified model class. This supports :4.3 and :4.5: failure of adjustment is not the general stopping rule, and a negative algorithmic result requires the matching completeness conditions. The result is not a universal certificate for cyclic, selected or otherwise differently specified models. Source.

Several available laws rather than an invented joint law. Tikka, Hyttinen and Karvanen, Causal Effect Identification from Multiple Incomplete Data Sources: A General Search-based Approach (2021; arXiv v5, 27 August 2021), §§2–3.4, develops do-search by retaining known distributions and deriving new ones with justified rules. This is adopted in :4.2–:4.3 when one adjustment formula or one complete observational law does not fit the information actually available. Its broader search costs more than a sufficient specialized derivation; its negative output has only the completeness scope established for the problem being solved. Read version.

Current extension of a familiar sufficient criterion. Wu and Robeva, Generalization of Pearl’s Front-Door Criterion (arXiv v1, 16 April 2026), §3, gives weaker sufficient graphical conditions for the same front-door functional and worked derivations outside the classical criterion. This reinforces the broader derivation route in :4.3 instead of rejecting an effect merely because a familiar criterion fails. The simple case in :5.2 needs only the classical sufficient conditions. The newer result concerns a particular functional in its stated graphical setting; it neither removes the need for subject justification nor supplies a complete test for every causal target. Read version.

Population scope is part of identification. Dahabreh and colleagues, Generalizing causal inferences from individuals in randomized trials to all trial-eligible individuals (2019; arXiv v2, 29 October 2019), §§2–4, separates within-trial exchangeability from the conditional-mean and participation assumptions required for generalization. That distinction is adapted in :4.4 and :5.3: the target mixture can differ even when the experimental comparisons are valid. Its particular nested-trial setup is not presumed for arbitrary selected records. Read version.

A local effect under a binary encouragement. The serious alternative to :4.3.1 is reporting the randomized offer’s effect, or seeking a population effect of actual use under additional assumptions. Adopt the narrower response-group derivation when its local target serves the receiving use: it needs no fully specified outcome mechanism, but does require substantive exclusion and monotonicity. The historical Angrist–Imbens–Rubin argument, 1996, §§2–5 supplies that identification result, not a complete modern inference method for weak instruments. The calculation and two compatible populations in :5.4 expose what the local result leaves undetermined. Reopen when assignment can affect outcomes directly, response types change, or the receiving population or target differs.

Reopen the chosen derivation when the target, available laws, causal exclusions, population bridge or support changes. Consider a different identification method when it answers the same question under more defensible assumptions or with materially less effort; source recency alone does not require replacing an already sufficient argument.

MMP.15:12 - Relations

• C.28 supplies the causal-use question and the distinction between identification, estimation and realizability. C.28.MR supplies intervention semantics and calculations within a specified causal model. C.28.CM constructs missing causal relations and material alternatives before their identifying implications are assessed.
• C.16.IR supplies compatible-case and sufficient-target reasoning; this pattern constructs the causal identification expressions and ambiguity witnesses.
• MMP.7 supplies the observation law, including selection and missingness. MMP.11 supplies an explicit model family when its restrictions are needed.
• MMP.12 handles inverse ambiguity and justified regularization. A restriction used here remains an added causal or response assumption, not new evidence.
• MMP.13 constructs inference for the identified expression or explicitly assumption-dependent target. MMP.14 investigates failed model predictions; observational checks alone need not distinguish observationally equivalent causal models.
• MMP.19 constructs and bounds history-conditioned, joint and nested counterfactual comparisons; an intervention mean can remain sufficient for a choice based on expected outcome.
• MMP.16 addresses a chosen need for additional observation design. C.11.DUA governs whether that work is worthwhile.
• Computational Thinking, including CMP.8/CMP.9 where their numerical methods apply, obtains numerical values without supplying the missing causal argument. Subject methods justify and realize the intervention and the asserted invariances.

MMP.15:End

MMP.19 - Construct and Bound a Mathematical Counterfactual Comparison under a Causal Model

Type: Method pattern Status: Stable Normativity: Normative unless marked informative

MMP.19:1 - Problem frame

Use this pattern when the question concerns the same case under another action, a joint comparison of its possible responses, or a specified pathway through which a change would act. A population average may be available while the case-specific or pathway question remains unanswered.

Start with the intended use, the observations about the case or population, and a causal account of the relevant mechanisms. Construct what remains common across the alternatives, carry factual information into that construction and calculate the consequence or range it supports. A useful first result can be two compatible models that give different answers: the supplied information then leaves this distinction unresolved.

The mathematical prerequisites are function evaluation and the probability operations used in the chosen construction. A qualified contributor can supply them. The meaning of the variables, observations and interventions still needs subject knowledge. Small finite cases can be calculated by hand.

For a population intervention effect, use MMP.15 and C.28.MR directly. A choice maximizing expected realized outcome under each action can also need only those intervention distributions, conditional on the information available before acting. Constructing a joint counterfactual law is useful when the receiving question actually depends on it.

MMP.19:2 - Problem

An intervention distribution describes results across the selected population. Information about an observed case can select different underlying conditions. Reusing the population distribution then answers another question.

There is also a difference between knowing each alternative’s distribution and knowing how their results correspond for the same unit. Two causal accounts can agree on every available observation and intervention yet disagree about whether this unit would have succeeded under both actions. Independent simulation of the two distributions silently chooses one such correspondence.

A pathway question adds another dependency: a variable may be set to the value it would have had under a different action. Substituting its population mean can erase the case relation that the question needs.

MMP.19:3 - Forces

MMP.19:4 - Solution

Specify the comparison → construct the common case → use its factual information → evaluate the alternative mechanisms → identify or bound the target → return the consequence to use.

MMP.19:4.1 - State what the comparison must answer

Name the unit, relevant time and population. State the action or mechanism alternatives and what would count as their outcomes. C.28 distinguishes the causal-use question; C.28.MR supplies the meaning of mechanism replacement.

Distinguish the targets that the question could need. Common examples are a mean under an intervention, the response under another action given what happened to this case, and a joint property such as succeeding under one action while failing under another. For a pathway question, name which mechanism’s input is to be changed and which other route remains as specified.

Recover what the recipient will do with the answer. A future choice based on expected outcome and action cost can depend only on each action’s marginal distribution. A probability of benefit, an explanation of an observed result or a nested pathway contrast can depend on more. C.11 and MMP.8 retain the decision criterion; the mathematical construction must answer that criterion rather than substituting another one.

MMP.19:4.2 - Construct one common basis for the alternatives

In a structural model, write the relevant variables as functions of their parents and underlying inputs U. Recover the joint law p(u), given input values, or a set of admissible input laws and mechanisms. U can include shared disturbances and persistent characteristics; its meaning depends on the subject account.

For each action a, let Y_a(u) denote the output of the modified model at the same underlying input u. A response-type representation can instead give a joint vector such as (Y_0,Y_1) directly. Its joint distribution states which responses belong to the same modeled unit. Having the two marginal distributions alone does not supply this dependence.

Keep common conditions common, and identify which changes the action itself causes. Two runs with independently drawn U describe different modeled cases. Reusing a random seed is a numerical way to implement a specified coupling; the seed does not justify that coupling in the subject. Repeating work later with changed conditions likewise needs an account of which inputs persist and which are newly drawn.

Use a model with well-defined responses for the requested inputs. In the finite acyclic case, evaluation in dependency order suffices. Continuous, cyclic or multiple-solution models require their own existence and solution-selection conditions; a solver’s returned trace does not settle an unspecified response relation.

MMP.19:4.3 - Infer case conditions from the factual observations

Let e denote the available factual record. Use the model and recording procedure that produced it, including the actual action regime. MMP.7 constructs the recording law; MMP.13 supplies the needed conditioning.

For a finite supplied input law and record likelihood k(e given u), form

w(u given e) = k(e given u) p(u) / sum_v k(e given v) p(v).

The denominator must be positive. An exact deterministic record has a likelihood of one for compatible inputs and zero for incompatible ones. A noisy or selected record needs its actual likelihood. An individually observed continuous value uses the appropriate conditional distribution or density, not division by the probability of a zero-probability point.

This operation uses the factual mechanisms. Conditioning after replacing them can select a different set of cases. If the supplied account cannot produce the record, return the conflict; changing an assumed mechanism or observation model is a separate justified repair.

A probability law need not be invented when only possible inputs are supported. Retain the inputs compatible with the record and derive the range of responses they permit. If there is no factual conditioning in the query, use the selected population law or set directly.

MMP.19:4.4 - Evaluate the compared responses with that common basis

Replace the selected mechanisms using C.28.MR. At each retained u, evaluate all the responses needed by the question before averaging. For a finite conditional comparison,

P(Y_a=y, Y_b=z given e)
    = sum_u w(u given e) 1[Y_a(u)=y and Y_b(u)=z].

Here the indicator is one when both conditions hold. A single response or expected difference uses the corresponding function inside the same sum. The shared u preserves the modeled relation between alternatives. Use integration when the model supports a continuous version of this operation.

For a nested response Y_(a,M_b)(u), first compute the intermediate value M_b(u) under b. In the second model set A to a and M to that computed value for the same u, then evaluate Y. Averaging M_b before this replacement generally changes the target. The construction defines a model consequence even when no available physical procedure can jointly realize every term; C.28:4.5 governs a separate claim about obtaining samples.

For a pathway decomposition, state both component contrasts and verify that their sum is the intended total. Interactions can make another choice of reference mechanism give different components. A component’s numerical size does not by itself establish the adequacy of the explanation or the feasibility of a proposed physical intervention.

MMP.19:4.5 - Determine what the available information fixes

A fully supplied model gives a consequence conditional on that model. Identification asks whether all admissible models agreeing with the available information give the same target. Use MMP.15’s distinction between a proof of ambiguity and an unfinished search on the counterfactual construction now specified.

Two compatible models with different target values prove non-identification under their shared assumptions and available laws. More samples from those same laws cannot distinguish them. A narrower quantity can nevertheless be identified.

For a finite response-type construction, assign a nonnegative mass to each admitted type, with total mass one. Express the known marginal, joint and regime laws as constraints on those masses. Add a structural restriction only when the causal account supports it. Minimize and maximize the target over the compatible masses; for a conditional probability retain its conditioning denominator. MMP.10 helps formulate the constraints and an appropriate computational method can solve them.

Call the resulting bounds tight only when the construction establishes that no smaller range follows and that its extremes are attainable, or specifies unattained limiting extremes. A numerical search that finds two values supplies witnesses, not necessarily the full range. Uncertainty from estimating an input law with finite data is another layer, handled by MMP.13; it is not the same as ambiguity remaining even when that law is known.

MMP.19:4.6 - Use a sufficient consequence and reopen the affected assumption

Return the target’s meaning, the supported value or range, and the assumptions that determine its use. The calculation can already carry these facts; a separate record is needed only by a receiving use.

If every compatible answer supports the same sufficient action or explanation, use that result. If the distinction matters, identify what would change it: a defensible restriction on mechanisms, informative existing records, a feasible different observation, or a different question. MMP.16 and C.11.DUA compare the value and burden of obtaining that contribution. Unavailable evidence can leave a bounded answer and a choice under uncertainty.

A revised recording rule changes the factual conditioning. A revised intervention changes the response functions. A changed criterion can make joint dependence irrelevant. Recalculate the affected contribution while retaining the rest.

During an investigation, this construction can perform its mathematical-modeling contribution while conditioning and mechanism evaluation supply constituent Methods. The investigator or subject specialist supplies the case and causal interpretation; a mathematical or AI collaborator may supply the calculation. Availability of one contribution leaves the others to be obtained where the work needs them.

MMP.19:5 - Archetypal Grounding

These are constructed cases with supplied mechanisms or exact probability laws. They demonstrate the mathematical operations and limits, not empirical effectiveness.

MMP.19:5.1 - Change the route for an already observed processing case

A routing model gives latency in ticks as Y=2A+U. Route A=1 adds two ticks; residual latency U is 0 or 1 with equal probability. The recorded case used A=1 and had Y=3.

Under the factual equation only U=1 is compatible. Keeping that input and replacing the route by A=0 gives Y_0=1. Drawing a fresh U from the population instead gives mean latency 1/2. That is a new-case mean, not the requested alternative for the observed case.

If the record is noisy, use its likelihood rather than selecting U with certainty. If an exact Y=4 is reported under the unchanged model, neither admitted U is compatible; return the conflict instead of calculating a posterior with zero denominator. Whether the same residual condition would persist under a real route change is a subject premise of this model.

MMP.19:5.2 - Expose a coupling that experiments do not determine

Let a randomized binary action A be independent of the response type U=(Y_0,Y_1). Consider two possible type distributions:

Each action succeeds with probability 1/2 in both models. With a fair random assignment, all four observed (A,Y) combinations have probability 1/4 in each model.

For a case observed with A=1,Y=1, only types (0,1) and (1,1) remain. In S, the probability that it would also succeed under A=0 is (3/8)/(1/2)=3/4. In T it is (1/8)/(1/2)=1/4. Both models fit the supplied observational and intervention laws, so these laws do not identify the answer.

For a future choice scored only by expected success, the actions tie at 1/2 under both models. If A=1 adds a positive cost and no other consequence, A=0 is sufficient for that criterion. Resolving the counterfactual ambiguity would not improve this choice.

MMP.19:5.3 - Bound benefit without inventing a joint law

Suppose binary success probabilities under actions 1 and 0 are p1=7/10 and p0=2/5. Let b=P(Y_0=0,Y_1=1), the probability of succeeding only under action 1. The four type masses must be

P(0,1)=b                 P(1,1)=p1-b
P(1,0)=p0-p1+b           P(0,0)=1-p0-b.

Nonnegativity gives max(0,p1-p0) <= b <= min(p1,1-p0), hence 3/10 <= b <= 3/5. At b=3/10 the masses in order (00,01,10,11) are (3/10,3/10,0,2/5); at b=3/5 they are (0,3/5,3/10,1/10). Both attain the supplied marginals, establishing the bounds for this unrestricted response-type class.

Now suppose assignment A is independent of the response pair (Y_0,Y_1), P(A=1)>0, and a case is observed with A=1,Y=1. The probability that this case would fail under action 0 is P(Y_0=0 given A=1,Y=1)=b/p1, hence between 3/7 and 6/7. Conditioning retains the types with Y_1=1; the independent assignment probability cancels from numerator and denominator. The same endpoint distributions attain these conditional bounds, since their denominator is the fixed positive p1=7/10.

The mean effect is p1-p0=3/10 in every compatible model. An expected-success criterion with an action-1 cost of 1/10 therefore has net gain 1/5 without identifying b. A criterion that explicitly requires b to exceed 2/5 remains unsettled by these bounds.

If subject knowledge warrants that action 1 never changes a success into failure, P(1,0)=0 fixes b=3/10. That is an additional monotonicity assumption; it was not learned from the two marginal probabilities.

MMP.19:5.4 - State which mediated contrast is being calculated

Take the supplied mechanisms M=A+U and Y=3A+2M+AM, with U=0 or 1. Compare actions 0 and 1 for the same u. The total change is 6+u.

First retain the mediator at M_0=u while changing A: Y_(1,M_0)-Y_(0,M_0)=3+u. Then change the mediator to M_1=1+u while retaining A=1: Y_(1,M_1)-Y_(1,M_0)=3. The components sum to 6+u.

Reversing that decomposition gives mediator change 2 at A=0, followed by direct change 4+u at M_1. These components also sum to 6+u. The interaction AM makes the two decompositions different. A request for “the part caused through M” must select its intended contrast. Knowing the total effect alone supplies neither decomposition; observing data that identify it is a further question.

MMP.19:6 - Bias-Annotation

The explicit examples use finite inputs and simple structural functions. Their transparency helps inspect a comparison but does not show that such a causal account is obtainable for every subject. Subject ambiguity can remain after a numerical method computes a definite answer.

The same-unit relation is part of the model. Its appropriateness in a physical, organizational or human setting depends on what is held common and what each action changes. A purely numerical pairing can be useful for variance reduction while lacking the interpretation required by the causal question.

MMP.19:7 - Conformance Checklist

• The target states the unit, alternatives, relevant factual information and receiving use.
• Common conditions and changed mechanisms have a subject interpretation.
• Factual conditioning uses the factual mechanism and recording law.
• Joint or nested responses use the specified common-case dependence.
• A model-conditional calculation is distinguished from identification across admissible models.
• An ambiguity or bound is returned at its proved scope, with estimation uncertainty kept distinct.
• A sufficient action can finish without recovering an irrelevant joint law.

MMP.19:8 - Common Anti-Patterns and How to Avoid Them

MMP.19:9 - Consequences

The recipient obtains a comparison whose case relation and assumptions can be inspected, changed and used. A proved range exposes what remains unresolved without replacing the result by a generic request for more data.

Constructing a common causal account and solving its constraints has a cost. Its value depends on whether the joint, conditional or pathway distinction changes the explanation or decision. An intervention mean or conditional outcome prediction can remain the sufficient result.

MMP.19:10 - Architectural Rationale

The hard step is often the relation between calculations, rather than each calculation separately. Ordinary conditioning recovers case information; mechanism replacement constructs an alternative; the common input or response-type law connects them. Maintaining that relation permits a joint question and exposes assumptions invisible in separate marginal fits.

The same construction supports historical explanation, probabilities of benefit and nested pathways. Their targets remain distinct. In particular, a decision based on realized outcomes can use less structure than a question about how those outcomes would differ for the same unit.

MMP.19:11 - SoTA-Echoing

Correa and Bareinboim (2025), Definitions 1.1–1.3 and §2.1.1, supply the shared-input and nested-response semantics used in :4.2–:4.4. The finite examples here use those definitions; the richer graphical calculus has further conditions and operations.

Mueller and Pearl (2023) separate average effects from probabilities of benefit or harm. Dawid and Senn (2025, v2), §§2–4, supply a serious alternative: intervention distributions suffice for maximizing expected realized outcomes under the stated information and constraints. Adopt the target distinction in :4.1/:4.6. In :5.2 the same two models require a joint account for the historical question but no such recovery for the stated future choice. The changed question justifies the more elaborate construction.

Raghavan and Bareinboim (2026, preprint), §§2 and 4–5, distinguish identification, partial identification and physically realizable counterfactual sampling. Adapt those distinctions in :4.4–:4.5. Their available-action, finite recursive and positivity assumptions limit the algorithmic results. Establishing a model consequence leaves its physical sampling conditions to be established separately when sampling is needed.

Reopen the selected construction when a supported additional mechanism or information source changes the compatible comparisons, or when another method obtains the required answer with less unsupported structure or burden. The worked finite constructions establish their stated mathematical consequences; practical validity still depends on the subject account.

MMP.19:12 - Relations

• C.28 governs causal-use questions and support; C.28.MR constructs mechanism replacements and their consequences.
• MMP.7/.13 supplies the recording and inferential operations used to recover factual case information. MMP.15 identifies intervention effects from available laws and supplies the identification/ambiguity distinction reused here.
• MMP.10 formulates compatible response-type constraints. MMP.16 constructs a worthwhile distinguishing observation; MMP.14 repairs an account contradicted by appropriate records.
• MMP.8/.8.SD, C.11 and C.11.DUA connect the consequence to choice, continuation and proportionate further inquiry. EXD.1 recovers what connection an explanation must supply.

MMP.19:End

MMP.16 - Design Observations to Separate Mathematical Model Alternatives

Type: Method pattern Status: Stable Normativity: Normative within the stated use

MMP.16:1 - Problem frame

Use this pattern when several mathematical accounts remain possible and the next useful answer depends on a difference between them. You can choose which cases to observe, under which conditions, or how to record them. The difficulty is to find an attainable observation that exposes the relevant difference.

The desired result may be a better decision, a supported prediction, a distinction between proposed mechanisms, or a new research direction. For example, two response laws agree at all previously tried inputs but imply different behavior in the intended use. Repeating those inputs more accurately may leave the disagreement untouched.

This pattern constructs the mathematical design of an observation. It returns what to observe, what records the alternatives would produce, and how those records would change the answer. It can instead return a useful limitation: the available observation choices leave the needed distinction unresolved. The subject practice supplies feasible access and performs the observation; PHY.9 and PHY.10 do that work for a physical readout and test.

Start with the receiving question, the alternative models and the observation law from MMP.7. Set-valued reasoning needs functions and sets. Probabilistic design also needs conditional probability and the inferential meaning chosen in MMP.13. An analyst can supply that mathematics while another participant supplies the observing conditions and interprets the result.

Use the present answer directly when the remaining alternatives make no consequential difference. A new observation is only one possible response to uncertainty. C.11.DUA supplies the comparison with using what is already known, changing the claim or taking another feasible step. A routine prescribed measurement with an already sufficient design needs no new design study.

MMP.16:2 - Problem

A model can predict a large difference that the actual record cannot show. An instrument may saturate, a log may aggregate away timing, or an unknown offset may account for the apparent difference. Conversely, overlapping distributions can still support a useful distinction; requiring one observation to identify the model without error can reject worthwhile designs.

An observation can also be highly informative about something irrelevant to the requested result. Choosing a design by parameter uncertainty, fit or model label can therefore direct effort away from the question that made the modeling useful.

The problem is to construct a feasible observation whose possible records support a consequential comparison, with the uncertainty and cost that this use can tolerate.

MMP.16:3 - Forces

MMP.16:4 - Solution

State the disagreement that matters. Derive the records each alternative can produce under feasible observation choices. Construct a comparison of those records that improves the receiving answer, and compare the attainable benefit with the work involved. Obtain the selected observation only when that comparison supports it; otherwise use the present result or change the question or access.

MMP.16:4.1 - Locate the consequential disagreement

Write the answer that would differ. It may be a threshold response, a predicted event, an intervention consequence or which conjecture to develop next. For a numerical target, write it as q(h,theta), where h selects a model and theta contains its remaining unknowns. Different theta within one model can matter as much as different model labels.

Recover what the present observations and assumptions leave possible. MMP.11 constructs the family of models; MMP.12 exposes a recovery ambiguity; MMP.13 supplies probabilistic conclusions when used. Group equivalent parameterizations by the behavior relevant to this question.

Ask which remaining disagreements change the intended answer or a worthwhile later inquiry. If all retained alternatives already support the same sufficient answer, return it.

MMP.16:4.2 - Construct the law of the obtainable records

Let d denote a feasible design: the selected inputs or cases, preparation, observing times and recording procedure. For each alternative derive either:

• a set R(h,theta,d) of possible records under its bounded uncertainties; or
• a probability law P(Y | h,theta,d) for the records Y.

The record includes what the procedure would actually retain. Compose the modeled response with the selection, measurement, censoring, rounding or aggregation operation. If an intervention changes the subject, derive its response under that intervention; an observational association alone supplies no such law.

Keep unknowns shared across readings shared. An unknown calibration offset cannot take one arbitrary value for the first reading and an unrelated value for the next if the same offset governs both. A repeated observation can reduce independent noise while leaving that common ambiguity intact.

A probabilistic comparison integrates unknowns only under a supplied probability law. Otherwise retain them as conditional possibilities or compare performance across their admitted range. Choosing a convenient nuisance value separately for each model can make a design look more discriminating than it is.

For an adaptive design, later observation choices are functions of records already available. Derive their joint law with that dependence. A sequence of fixed-design calculations does not by itself describe an outcome-dependent stopping or sampling rule.

MMP.16:4.3 - Find what a design can distinguish

First seek a simple consequential contrast. For a set-valued model, take the union of R(h,theta,d) over the remaining possible theta. Disjoint unions for two alternatives give a separating observation under those assumptions. If the unions overlap, identify records that would settle the needed distinction and records that would leave it open.

Equal sets alone do not establish equal statistical information: probability laws can weight the same possible records differently. In a probabilistic model, compare the full record laws or a statistic whose retained information suffices for the requested result. Distinct means are one possible contrast, not a universal criterion.

An impossibility conclusion needs its scope. Two alternatives that give the same record law under every currently feasible design, yet different required answers, demonstrate a distinction unavailable through those designs. More repetitions under the same uninformative access do not resolve it. Changing the observation type, access or target may do so. Failure of a numerical search to find a good design establishes only that search result.

A narrower target can remain obtainable. Suppose all feasible records leave the individual parameters unresolved, but every compatible parameter pair gives the same total response. Return the total when it answers the work question. Recovering each parameter is then unnecessary.

MMP.16:4.4 - Construct the design criterion from the receiving use

For a required distinction with controlled statistical error, construct a rule T(Y) that returns the answer or an unresolved result. For example, let two specified hypotheses have record densities p0(y;d) and p1(y;d) relative to the same measure. For a chosen threshold c, the rule selects H1 on the region A={y: p1(y;d)>c*p0(y;d)} and H0 otherwise. Calculate P0(A), the chance of selecting H1 under H0, and P1(A-complement), the opposite error. Compare designs and thresholds that meet the required error bound. For a composite hypothesis, the claimed protection across its parameter range requires controlling the error across that range, rather than only at a fitted value.

Other decision rules can retain an unresolved answer when that is useful. Their comparison likewise follows from their record laws and the error consequences the use requires.

When probability, actions and losses are appropriate, let pi be the current joint law of the unknowns, a an available action, and L(a,h,theta) its loss. For a finite set of action choices and an observation that changes information alone, compare:

R0 = min_a E_pi[L(a,h,theta)]

R(d) = E_Y[min_a E[L(a,h,theta) | Y,d]]

value of sample information = R0 - R(d).

The inner choice uses the observed record; the outer expectation averages records that are still unknown when the design is chosen. In this formulation the recipient can ignore the record and retain the old action, so R(d) cannot exceed R0 under the same model. Subtracting the full cost of obtaining and using the information can still make the proposal unattractive. :5.2 carries out the calculation.

If the experiment itself changes the state, available actions or their consequences, include those effects in the decision model. The simple information-only comparison above is then insufficient. MMP.8.SD constructs the continuing state, information and consequence model.

An information criterion is another branch. For a selected unknown Q, expected information gain is the mutual information I(Q;Y | d) under the supplied joint law. Choosing Q as the model label, all parameters or a wanted prediction defines different design problems. Use this criterion when resolving that uncertainty serves the stated inquiry. It does not measure every practical consequence of the observation.

The comparison need not be a probability calculation. A guaranteed separation, an ordinal improvement, or a change in attainable answers can suffice. Use the existing FPF choice and portfolio methods when several gains and burdens remain incomparable. This pattern supplies the modeled observation consequences, not a new general system for valuing research.

MMP.16:4.5 - Construct and compare attainable designs

Use the contrast to generate alternatives: observe where responses differ, resolve an omitted component, vary an input independently, measure at a different scale or time, or change how the record is made. Derive the resulting record laws before optimizing a convenient proxy.

For a small set of designs, calculate the selected criterion directly. For a large set, use the applicable search or numerical method from Computational Thinking. Count the cost of evaluating a design as part of the work. If approximate criteria cannot reliably order close candidates, return that uncertainty, retain several designs, or refine the calculation where a changed ranking matters.

Compare against using current information. C.11.DUA governs the attainable contribution, delay, displaced work and any disputed evidence demand. When an information-only design has a bound on its possible benefit below its cost, that bound can end the comparison without computing its criterion more accurately.

For a sequence of observations, decide whether the immediate comparison represents the intended horizon. An apparently uninformative first step may enable a later separating observation. Construct that continuation with MMP.8.SD when it changes the choice; do not require a full sequential optimization for a sufficient one-step design.

MMP.16:4.6 - Use the result and reopen the implicated assumption

Return enough for the observation to be performed and interpreted: the selected conditions, records to retain, comparison rule, and the consequence of an unresolved or conflicting result. A design calculation remains a prediction about possible records; it is not an observation already obtained.

After observing, use the inference and comparison that match the actual procedure. If access, stopping or recording changed, revise the affected law before interpreting the result. For example, replacing a numerical readout by a threshold alarm changes the available distinction.

Model discrimination compares the alternatives supplied. If every alternative fails to explain an important record, MMP.14 returns to the relevant subject or observation assumption. Selecting the least poor alternative can support a limited approximation, but its winning score alone does not establish adequacy.

Where the intended use covers different conditions from the selected observations, carry that difference into the receiving prediction. Recent work on active learning under model misspecification shows why concentrating observations for parameter information can worsen prediction elsewhere. Compare alternative observation regions or model families when this vulnerability can change the design or receiving prediction.

MMP.16:5 - Archetypal Grounding

The cases use stipulated models and costs so that the design can be reconstructed. They demonstrate the method, not reports of performed experiments.

MMP.16:5.1 - The largest response difference disappears in the readout

A team has two candidate response laws for a calibrated device assumed valid over the input range 0 to 2:

• H1: y=x;
• H2: y=x^2.

Both agree at the previously inspected inputs 0 and 1. The team needs to know whether the modeled response at x=2 is below or above 3. The laws answer differently: 2 and 4. Their applicability across the range is an assumption supplied for this case.

With a numerical readout r=y+e and a known error bound -0.1 <= e <= 0.1, the design x=2 gives:

Now recover a missed feature of the actual instrument: it saturates at 1. Its record is r=min(1,y+e). At x=2 both alternatives always record 1. The large latent response difference gives no distinction at all.

Changing the input to x=0.5 yields:

This design separates the supplied alternatives without replacing the instrument. A record 0.27 retains H2; a record 0.52 retains H1. A record 0.37 fits neither under the given error bound, so it reopens the response or observing assumptions rather than forcing a label.

The result at x=0.5 supports the answer at x=2 through the supplied response families. If those families were justified only up to x=1, this design would not establish the requested extrapolation. The next work would concern that range extension or access to a suitable direct observation.

If the only available inputs were 0 and 1 and the record error law were the same under both accounts, the available designs would not separate them. That conclusion concerns the stated access; it does not say the response question is unanswerable under every possible instrument.

MMP.16:5.2 - A useful signal is still too costly

A service team must choose one of two recovery procedures. It models two possible failure modes H0 and H1 with current probabilities 0.5 each. The loss is remaining recovery time, in hours:

With current information, expected losses are 5 for A and 2 for B. The team chooses B.

A proposed diagnostic produces either “+” or “-”. Its supplied law is P(+ | H1)=0.8 and P(+ | H0)=0.2. Each signal has marginal probability 0.5. Bayes conditioning gives P(H1 | +)=0.8 and P(H1 | -)=0.2.

After “+”, A has expected loss 8 and B has 0.8, so the team chooses B. After “-”, A has expected loss 2 and B has 3.2, so it chooses A. Expected loss with the signal, excluding diagnostic cost, is therefore:

R(d) = 0.5*0.8 + 0.5*2 = 1.4 hours.
R0 - R(d) = 2 - 1.4 = 0.6 hours.

If obtaining, interpreting and waiting for the diagnostic adds 0.7 hours to recovery, its total expected loss is 2.1 hours; using current information is better under these conditions. At an all-inclusive cost of 0.2 hours, the diagnostic gives 1.6 hours and improves the choice. The example assumes the diagnostic does not change the failure mode or the two procedures.

A different test might reveal a device identifier perfectly. If that identifier is independent of the failure mode and losses in this model, its information gain about the identifier does not reduce this recovery loss. Selecting all available uncertainty as the target would misdirect the design.

Now change the current probability of H1 to 0.95. The “-” posterior is 0.19/(0.19+0.04), approximately 0.826, and the “+” posterior is 0.76/(0.76+0.01), approximately 0.987. B remains the better procedure after either result. The diagnostic still changes beliefs about the mode, but its sample-information value for this particular choice is zero.

A later method-development investigation could use the same signal for another target, such as learning how failure modes arise. That research value needs its own question and horizon; the zero above applies to the specified recovery choice.

MMP.16:6 - Bias-Annotation

The worked cases make the candidate models and observing conditions unusually explicit. In practice, deriving the record law can be the hardest contribution. The mathematical design must retain uncertainty about that law rather than quietly treating a convenient simulator as the subject.

A small named model set also favors choosing a winner. Keep the possibility that all proposed accounts fail the receiving use. Conversely, do not turn that possibility into a demand for an unlimited search for alternatives: pursue it when a discrepancy or consequential vulnerability makes it useful.

MMP.16:7 - Conformance Checklist

A constructed design supports its intended use when:

• the remaining disagreement and the answer it could change are identifiable;
• its predicted records follow from the alternatives and the actual selection and recording procedure;
• shared unknowns and adaptive choices retain their dependencies;
• the claimed separation, error property or expected improvement follows from a reconstructible comparison;
• any criterion based on information names the unknown being learned and its receiving use;
• feasibility and burden are compared at the scope needed for the decision, using present information as an available alternative;
• the result explains how the selected records change the answer and when an incompatible record returns to the model or observation assumptions.

These conditions concern the mathematical design and its stated assumptions. They do not establish that the observation was performed, that its subject assumptions hold, or that access is authorized. Stronger assurance belongs to the actual intended use.

MMP.16:8 - Common Anti-Patterns and How to Avoid Them

MMP.16:9 - Consequences

The next observation becomes a constructed part of modeling: a specified change in available information with a stated consequence. A no-separation result can redirect effort from repeated measurement to a different readout, access route or question.

Good designs depend on assumptions about records and their use. A computed optimum can change when the observer, cost, model family or research horizon changes. For expensive searches, a sufficient feasible design may be more useful than a more precisely optimized proposal.

In methodology, the same construction helps compare alternative accounts of a working method. Choose a case or observation that makes their consequential disagreement visible. Human and automated procedures can both be studied this way; the subject method supplies what can be changed and observed.

MMP.16:10 - Architectural Rationale

The general contribution is the construction of discriminating observations from models and their receiving use. It is separate from performing a physical test, deriving an estimator after observations, or choosing among all possible projects.

Separating the predicted response from its recording prevents a mathematical contrast from being mistaken for an observable one. Separating information from decision consequence lets exploratory research retain its own value while avoiding an automatic demand for more evidence in ordinary work.

The deterministic and probabilistic branches share that construction. Their guarantees differ, so the pattern does not reduce every use to one statistical test or one information metric. Existing FPF choice, resource and improvement methods receive the consequences produced here.

MMP.16:11 - SoTA-Echoing

Design according to its goal. Huan, Jagalur and Marzouk, Optimal experimental design: Formulations and computations (Acta Numerica, 2024), §2.2 and the goal-oriented and model-discrimination formulations, distinguish information targets and decision utilities. The adopted contribution is the explicit selection of what an observation should improve in :4.4. A design maximizing information about every parameter is a serious default when that is the actual aim; it is replaced when the receiving target differs. Numerical optimization methods are delegated to the applicable computational construction. Source.

Information that can change a decision. Heath and colleagues, Simulating Study Data to Support Expected Value of Sample Information Calculations: A Tutorial (2022), “Background and Notation,” gives the pre-observation comparison of decisions after possible records. Its loss-form equivalent is used in :4.4 and the original recovery example in :5.2. The model’s costs and consequences must describe the receiving work; the source’s health-economic conventions are not generalized into mandatory monetary valuation. Source.

An informative design can misdirect learning. Tang, Sloman and Kaski, Representative, Informative, and De-Amplifying: Requirements for Robust Bayesian Active Learning under Model Misspecification (arXiv v2, 2026), §§2–4, studies prediction under a fixed, potentially inadequate model family. Its analysis separates approximation error, estimation error and their interaction under the intended input distribution. This motivates the explicit receiving conditions in :4.6. Their proposed acquisition rule is not a universal remedy: its assumptions and quantities need their own justification before use. The adopted general move is to compare consequential model inadequacy and observation placement, rather than assuming that higher information gain implies better prediction. Read version.

These sources develop statistical branches. The bounded-error construction in :5.1 is an elementary set-valued derivation; it needs no prior distribution. Reopen the selected design when the target, record law, feasible access, receiving population or substantive model alternatives change. A newer optimization technique matters when it improves that same construction at worthwhile effort.

MMP.16:12 - Relations

• MMP.7 constructs the law of what is recorded; MMP.11 constructs the admitted model family.
• MMP.12 exposes recovery ambiguities; MMP.13 supplies inference and uncertainty claims that match the chosen observation procedure.
• MMP.14 investigates failed predictions and returns to the implicated subject or observation assumption.
• MMP.15 supplies causal identification when the target is an intervention effect. A new association does not by itself identify that effect.
• MMP.8 formulates the receiving choice; MMP.8.SD constructs a longer observation-and-action sequence when continuation changes the design.
• C.11.DUA appraises an evidence demand and its attainable contribution; C.11 and the applicable portfolio methods compare attainable gains with resources and other consequences.
• C.16.IR supports a sufficient answer despite unresolved distinctions. B.5.TC relates a test to the theoretical comparison it is meant to change.
• PHY.9 constructs a physical readout and PHY.10 a physical test. Other subject practices realize the corresponding access and observation.
• Computational Thinking supplies the selected search, sampling or approximation procedure. MMP.17 consumes an informative-case design when constructing a surrogate.

MMP.16:End