MCMC Generated Bayesian Credible Intervals in Hierarchical Models

Under a Bayesian approach to a hierarchical model interest often lies in summarizing the posterior distribution of a parameter in order to enable easy comparison across prior structures This can be done for example through quantile or interval estimation When using an MCMC algorithm such as the Gibbs sampler to generate a sample from the posterior of interest calcu lations are often easier when done on a per iteration basis with the nal result then taken from a combination across iterations this is often called Rao Blackwellization Such an approach is not yet used in the calculation of credible regions We examine the performance of a weighted average estimator of the endpoints of a credible region and compare it to other alternatives including a na ve average estimator an order statistics estimator and an estimator based on density estimation We obtain theorems showing when there is convergence to the true interval and discuss Central Limit Theorems for these estimators An animal epidemiology example is used to examine the estimators numerical behavior