Robust Bayes and empirical Bayes analysis with \(\epsilon\)-contaminated priors. (English) Zbl 0602.62004

Berger, James; Berliner, L. Mark

Ann. Stat. 14, 461-486 (1986).

A robust Bayesian viewpoint assumes only that subjective information can be quantified in terms of a class \(\Gamma\) of possible distributions. The goal is then to make inferences which are relatively insensitive to deviations as the prior distribution varies over \(\Gamma\). The authors show that it is possible to work with complex classes of priors and indicate mathematical techniques for doing so. Both intuitive and calculational reasons are pointed out for approaching robustness through consideration of \(\epsilon\)-contamination classes, i.e., classes of prior distributions \(\Gamma =\{\pi:\pi =(1-\epsilon)\pi_ 0+\epsilon q\), \(q\in Q\}\), where \(0\leq \epsilon \leq 1\) is given, \(\pi_ 0\) is a particular prior distribution, and Q is a class of contaminations.
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 \(\epsilon\)-contamination class. The second issue is that of selecting, in a data dependent fashion, a good prior distribution from the \(\epsilon\)-contamination class, and using this prior in the subsequent analysis. Finally, applications to empirical Bayes analysis are discussed.

Reviewer: J.Melamed

MSC:

62A01	Foundations and philosophical topics in statistics
62F15	Bayesian inference

Keywords:

symmetric, unimodal contaminations; type-II maximum likelihood prior; epsilon contamination; selecting a prior distribution; determining the range of posterior probabilities; complex classes of priors; robustness; robust Bayesian analysis; empirical Bayes