Open Access
June, 1986 Robust Bayes and Empirical Bayes Analysis with $_\epsilon$-Contaminated Priors
James Berger, L. Mark Berliner
Ann. Statist. 14(2): 461-486 (June, 1986). DOI: 10.1214/aos/1176349933

Abstract

For Bayesian analysis, an attractive method of modelling uncertainty in the prior distribution is through use of $\varepsilon$-contamination classes, i.e., classes of distributions which have the form $\pi = (1 - \varepsilon)\pi_0 + \varepsilon q, \pi_0$ being the base elicited prior, $q$ being a "contamination," and $\varepsilon$ reflecting the amount of error in $\pi_0$ that is deemed possible. Classes of contaminations that are considered include (i) all possible contaminations, (ii) all symmetric, unimodal contaminations, and (iii) all contaminations such that $\pi$ is unimodal. Two issues in robust Bayesian analysis are studied. The first is that of determining the range of posterior probabilities of a set as $\pi$ ranges over the $\varepsilon$-contamination class. The second, more extensively studied, issue is that of selecting, in a data dependent fashion, a "good" prior distribution (the Type-II maximum likelihood prior) from the $\varepsilon$-contamination class, and using this prior in the subsequent analysis. Relationships and applications to empirical Bayes analysis are also discussed.

Citation

Download Citation

James Berger. L. Mark Berliner. "Robust Bayes and Empirical Bayes Analysis with $_\epsilon$-Contaminated Priors." Ann. Statist. 14 (2) 461 - 486, June, 1986. https://doi.org/10.1214/aos/1176349933

Information

Published: June, 1986
First available in Project Euclid: 12 April 2007

zbMATH: 0602.62004
MathSciNet: MR840509
Digital Object Identifier: 10.1214/aos/1176349933

Subjects:
Primary: 62A15
Secondary: 62F15

Keywords: $\epsilon$-contamination , classes of priors , Empirical Bayes , hierarchical priors , Robust Bayes , type II maximum likelihood

Rights: Copyright © 1986 Institute of Mathematical Statistics

Vol.14 • No. 2 • June, 1986
Back to Top