Ranges of Posterior Probabilities for Quasiunimodal Priors with Specified Quantiles

Abstract

Suppose several quantiles of the prior distribution for θ are specified or, equivalently, the prior probabilities of a partitioning collection of intervals {I_i } are given. In addition, suppose that the prior distribution is assumed to be unimodal. Rather than selecting a single prior distribution to perform a Bayesian analysis, it is of interest to consider the class of all prior distributions compatible with these inputs. For this class and unimodal likelihood functions, the ranges of the posterior probabilities of the I_i and the ranges of the posterior cdf at the specified prior quantiles were determined in Berger and O'Hagan (in press). Unfortunately, calculations with this class can be difficult. Here a similar, much more easily analyzed class of quasiunimodal prior distributions is considered and compared with other classes.

Key Words:

• Prior quantiles
• Bayesian robustness