Decision-theoretic justifications for Bayesian hypothesis testing using credible sets

In Bayesian statistics the precise point-null 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 decision-theoretic justification. We argue 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 a special case HPD sets, by studying a three-decision 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.