Ranges of Posterior Measures for Priors with Unimodal Contaminations

The Annals of Statistics · 1989 · Volume 17 · Issue 2

Abstract

We consider the problem of robustness or sensitivity of given Bayesian posterior criteria to specification of the prior distribution. Criteria considered include the posterior mean, variance and probability of a set (for credible regions and hypothesis testing). Uncertainty in an elicited prior, $\pi_0$, is modelled by an $\varepsilon$-contamination class $\Gamma = \{\pi = (1 - \varepsilon)\pi_0 + \varepsilon q, q \in Q\}$, where $\varepsilon$ reflects the amount of probabilistic uncertainty in $\pi_0$, and $Q$ is a class of allowable contaminations. For $Q = \{$all unimodal distributions$\}$ and $Q = \{\text{all symmetric unimodal distributions}\}$, we determine the ranges of the various posterior criteria as $\pi$ varies over $\Gamma$.

Keywords

$\varepsilon$-contamination classes of priors, 62C10, 62F15, 62F35, Bayesian robustness, posterior mean, posterior probability, posterior variance

Enable JavaScript for citation tools and additional paper interactions.