Hierarchical Model for Bayesian Hypothesis Testing

Abstract

A common problem in data analysis is testing a point hypothesis. For example, one might test whether or not the mean of a distribution is zero. We describe the standard Bayesian approach, which requires the choice of a width parameter to define a prior distribution for an alternative hypothesis. Standard practice is to choose this width parameter to be the least favorable choice for the original, point hypothesis. This approach, however, is ad-hoc and not informed by the data. We examine this practice in more detail than in previous work by constructing a hierarchical model, where the width parameter is an unknown value subject to inference. This yields a marginal posterior distribution for the width parameter. We then show that using the median of this marginal posterior distribution as the width parameter in the Bayesian comparison between the hypotheses gives results which are very similar to those obtained from the existing approaches. This gives a more rigorous and data-informed justification for the standard practice of choosing a width parameter which is least favorable to the point hypothesis.