The Annals of Statistics

Evaluation of formal posterior distributions via Markov chain arguments

Morris L. Eaton, James P. Hobert, Galin L. Jones, and Wen-Lin Lai

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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(‖θ2)  on the parameter space ℝp. Conditions on g for strong admissibility of the posterior are provided.

Article information

Source
Ann. Statist., Volume 36, Number 5 (2008), 2423-2452.

Dates
First available in Project Euclid: 13 October 2008

Permanent link to this document
https://projecteuclid.org/euclid.aos/1223908098

Digital Object Identifier
doi:10.1214/07-AOS542

Mathematical Reviews number (MathSciNet)
MR2458193

Zentralblatt MATH identifier
1274.62078

Subjects
Primary: 62C15: Admissibility
Secondary: 60J05: Discrete-time Markov processes on general state spaces

Keywords
Admissibility formal Bayes improper prior distribution multivariate normal distribution recurrence superharmonic function

Citation

Eaton, Morris L.; Hobert, James P.; Jones, Galin L.; Lai, Wen-Lin. Evaluation of formal posterior distributions via Markov chain arguments. Ann. Statist. 36 (2008), no. 5, 2423--2452. doi:10.1214/07-AOS542. https://projecteuclid.org/euclid.aos/1223908098


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