Decisiontheoretic justifications for Bayesian hypothesis testing using credible sets
Abstract
In Bayesian statistics the precise pointnull hypothesis $\theta=\theta_0$ can be tested by checking whether $\theta_0$ is contained in a credible set. This permits testing of $\theta=\theta_0$ without having to put prior probabilities on the hypotheses. While such inversions of credible sets have a long history in Bayesian inference, they have been criticised for lacking decisiontheoretic justification. We argue that these tests have many advantages over the standard Bayesian tests that use pointmass probabilities on the null hypothesis. We present a decisiontheoretic justification for the inversion of central credible intervals, and in a special case HPD sets, by studying a threedecision problem with directional conclusions. Interpreting the loss function used in the justification, we discuss when test based on credible sets are applicable. We then give some justifications for using credible sets when testing composite hypotheses, showing that tests based on credible sets coincide with standard tests in this setting.
 Publication:

arXiv eprints
 Pub Date:
 October 2012
 arXiv:
 arXiv:1210.1066
 Bibcode:
 2012arXiv1210.1066T
 Keywords:

 Mathematics  Statistics Theory
 EPrint:
 11 pages, 2 figures