Simultaneous Posterior Probability Statements From Monte Carlo Output

This article considers the problem of making simultaneous probability statements in multivariate inferential problems based on samples from a posterior distribution. The calculation of simultaneous credible bands is reviewed and—as an alternative—contour probabilities are proposed. These are defined as 1 minus the content of the highest posterior density region which just covers a certain point of interest. We discuss a Monte Carlo method to estimate contour probabilities and distinguish whether or not the functional form of the posterior density is available. In the latter case, an approach based on Rao-Blackwellization is proposed. We highlight that this new estimate has an important invariance property. We illustrate the performance of the different methods in three applications.