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.