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^Ttheta has posterior variance a^TV*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.