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RANGE OF THE POSTERIOR PROBABILITY

OF AN INTERVAL FOR PRIORS WITH

UNIMODALITY PRESERVING CONTAMINATIONS

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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 *epsilon*-contamination class
*Gamma* = {*pi* = (1-*epsilon*)*pi*_{0} + *epsilon*
*q*: *q* \in *Q*} is derived. Here, *pi*_{0} is the elicited prior which is
assumed unimodal, *epsilon* is the amount of uncertainty in *pi*_{0}, and *Q*
is the set of all probability densities *q* for which
*pi* = (1-*epsilon*)*pi*_{0} + *epsilon* *q* is unimodal with the same mode as that of
*pi*_{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-*epsilon*)*pi*_{0} except
in one interval (resp. two disjoint intervals) where it is constant.

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

**Source**
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