RANGE OF THE POSTERIOR PROBABILITY OF AN INTERVAL FOR PRIORS WITH UNIMODALITY PRESERVING CONTAMINATIONS

S. SIVAGANESAN

Department of Mathematical Sciences, University of Cincinnati, Old Chemistry Building (ML 25), Cincinnati, OH 45221-0025, U.S.A.

(Received February 21, 1991; revised February 27, 1992)

Abstract. Range of the posterior probability of an interval over the ϵ-con-tamination class Γ = {π = (1 − ϵ)π_{0} + ϵq : q ∈ Q} is derived. Here, π_{0} is the elicited prior which is assumed unimodal, ϵ is the amount of uncertainty in π_{0}, and Q is the set of all probability densities q for which π = (1 − ϵ)π_{0} + ϵq is unimodal with the same mode as that of π_{0}. We show that the sup (resp. inf) of the posterior probability of an interval is attained by a prior which is equal to (1 − ϵ)π_{0} except in one interval (resp. two disjoint intervals) where it is constant.

Key words and phrases: Posterior probability, unimodality preserving contaminations, Bayesian robustness.