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.