Matching Priors for Highest Posterior Density Regions

Abstract

Highest posterior density (HPD) regions are very popular with Bayesians. With a possibly multidimensional interest parameter θ, such a region is of the form

$$\{\tilde\theta: \pi (\tilde{\theta}|X) \geq K\}$$

, where $\pi (\tilde{\theta}|X)$ is the posterior density of θ, under a prior π(·), given the data X, and K depends on π(·) and X in addition to the chosen posterior credibility level. Clearly, by the Neyman-Pearson lemma, an HPD region has the smallest possible volume, given X, at a chosen level of credibility. In this chapter, we consider priors ensuring approximate frequentist validity of HPD regions with margin of error o(n ^-1 ), where n is the sample size. Priors of this kind are called matching priors for HPD regions or, briefly, HPD matching priors. They can be useful even when the interest parameter is multidimensional since HPD regions are well-defined in such situations.