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.