Bayesian Monte Carlo testing with one-dimensional measures of evidence

Abstract

Bayesian hypothesis testing procedures are constructed by means of test statistics which are functions of the posterior distribution. Usually, the whole sample vector is selected to form the sufficient empirical part of the posterior distribution. But, in certain problems, one may prefer to use well-established one-dimensional sufficient statistics in place of the sample vector. This paper introduces a Bayesian Monte Carlo procedure specially designed for such cases. It is shown that the performance of this new approach is arbitrarily close to the exact Bayesian test. In addition, for arbitrary desired precisions, we develop a theoretical rule of thumb for choosing the minimum number of Monte Carlo simulations. Surprisingly, does not depend on the shape of loss/cost functions when those are used to compound the test statistic. The method is illustrated for testing mean vectors in high-dimension and for detecting spatial clusters of diseases in aggregated maps.