2.5 The Pure Bayes (PB) Method of Credible Subgroups

The HPD and RCS methods leverage the frequentist properties of estimates of parameters and linear combinations of parameters to make frequentist coverage guarantees, but are conservative when considering posterior probabilities only. Exact credible subgroups may be obtained by replacing qF(1 − α, q, 2a) in equation (15) with some smaller value r². This yields a larger exclusive credible subgroup and a smaller inclusive credible subgroup.

Given a sample from the posterior of γ and a finite set C of points in the predictive covariate space, a Monte Carlo method estimates an appropriate value of r² via binary search:

Initialization: Set search bounds rL2=0 and rU2=qF(1−α,q,2a).

Set the working value for r to r^2=(rL2+rU2)/2.

Substitute $\widehat{r}$² for qF(1 − α, q, 2a) in (15) to produce a working subgroup pair ($\widehat{D}$, Ŝ).

Use the posterior sample of γ to produce a sample of B_γ and estimate $\widehat{p}$ = P_Bγ ($\widehat{D}$ ⊆ B_γ ⊆ Ŝ|y).

If $\widehat{p}$ > 1 − α set rU2=r^2, and if $\widehat{p}$ < 1 − α set rL2=r^2.

If $\widehat{p}$ is in [1 − α, 1 − α + ε), set r² = $\widehat{r}$² and end; otherwise go to (2).

When the set C or the posterior sample size is small, the algorithm may not reach the target precision for $\widehat{p}$, in which case the smallest $\widehat{p}$ > p may be taken.