Ranges of Posterior Measures for Priors with Unimodal Contaminations

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, π₀, is modelled by an ε-contamination class Γ = \π = (1 - ε)π₀ + ε q, q ∈ Q\, where ε reflects the amount of probabilistic uncertainty in π₀, and Q is a class of allowable contaminations. For Q = unimodal distributions\ and Q = symmetric unimodal distributions\, we determine the ranges of the various posterior criteria as π varies over Γ.