Evaluation of formal posterior distributions via Markov chain arguments
Morris L. Eaton, James P. Hobert, Galin L. Jones, and Wen-Lin Lai
Source: Ann. Statist. Volume 36, Number 5 (2008), 2423-2452.
Abstract
We consider evaluation of proper posterior distributions obtained from improper prior distributions. Our context is estimating a bounded function φ of a parameter when the loss is quadratic. If the posterior mean of φ is admissible for all bounded φ, the posterior is strongly admissible. We give sufficient conditions for strong admissibility. These conditions involve the recurrence of a Markov chain associated with the estimation problem. We develop general sufficient conditions for recurrence of general state space Markov chains that are also of independent interest. Our main example concerns the p-dimensional multivariate normal distribution with mean vector θ when the prior distribution has the form g(‖θ‖2) dθ on the parameter space ℝp. Conditions on g for strong admissibility of the posterior are provided.
Primary Subjects: 62C15
Secondary Subjects: 60J05
Keywords: Admissibility; formal Bayes; improper prior distribution; multivariate normal distribution; recurrence; superharmonic function
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Permanent link to this document: http://projecteuclid.org/euclid.aos/1223908098
Digital Object Identifier: doi:10.1214/07-AOS542
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