MMP.18:4.5 - Combine uncertainty without duplicating information

When components are probabilistic, construct a joint law for the common quantities and the component-specific quantities. Marginal distributions alone generally leave dependence unspecified. Obtain the needed dependence from the modeled mechanism, a conditional law or an explicit additional assumption.

For components sharing z, one possible factorization is:

p(z,u1,u2) = p(z) * p(u1 | z) * p(u2 | z).

It assumes conditional independence of u1 and u2 given z. If that assumption is unavailable, construct their joint conditional law or retain the unresolved dependence. A common random quantity is sampled or integrated once as that same quantity; two independent draws would describe a different model.

For two analyses using the same positive prior pi(z) and conditionally independent data D1,D2, their posteriors satisfy q1(z) proportional to pi(z)*L1(z) and q2(z) proportional to pi(z)*L2(z). The combined posterior is proportional to q1(z)*q2(z)/pi(z) on the common prior support. Multiplication without the division counts the prior twice.

If the analyses use overlapping records, first recover which observations and likelihood factors are shared. Dividing out a prior does not remove a duplicated observation. If component priors disagree, selecting a common prior or a pooling rule changes the model and needs a stated basis. MMP.7 supplies the record law and MMP.13 the resulting inference; a computational sampler obtains values from that law.

Point estimates can still be sufficient for a particular receiving use. To replace a distribution by a point, establish that the omitted uncertainty does not alter the required result at its chosen tolerance. Keep a consequential dependence when a nonlinear operation or tail probability needs it.