On decision-theoretic justifications for Bayesian hypothesis testing through 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 there are such justifications, by showing that inversion of central credible intervals can be motivated by studying a three-decision problem with directional conclusions, that the inversion of credible bounds are justified by a weighted 0-1 loss and by reviewing justifications for inverting HPD sets.