Range of the posterior probability of an interval for priors with unimodality preserving contaminations

Range of the posterior probability of an interval over the ε-contamination class Γ={π=(1−ε)π0+εq:qεQ} is derived. Here, π0 is the elicited prior which is assumed unimodal, ε is the amount of uncertainty in π0, andQ is the set of all probability densitiesq for which π=(1−ε)π0+εq is unimodal with the same mode as that of π0. 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)π0 except in one interval (resp. two disjoint intervals) where it is constant.

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