Decision-theoretic justifications for Bayesian hypothesis testing using credible sets

In Bayesian statistics the precise point-null hypothesis θ = θ 0 can be tested by checking whether θ0 is contained in a credible set. This permits testing of θ = θ 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 criticized for lacking decision-theoretic justification. ue that these tests have many advantages over the standard Bayesian tests that use point-mass probabilities on the null hypothesis. We present a decision-theoretic justification for the inversion of central credible intervals, and in special case HPD sets, by studying a three-decision problem with directional conclusions. Interpreting the loss function used in the justification, we discuss when tests based on credible sets are applicable. n 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.