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September, 1994 Sampling Models which Admit a Given General Exponential Family as a Conjugate Family of Priors
Shaul K. Bar-Lev, Peter Enis, Gerard Letac
Ann. Statist. 22(3): 1555-1586 (September, 1994). DOI: 10.1214/aos/1176325643


Let $\mathscr{K} = \{K_\lambda: \lambda \in \Lambda\}$ be a family of sampling distributions for the data $x$ on a sample space $\mathscr{X}$ which is indexed by a parameter $\lambda \in \Lambda,$ and let $\mathscr{F}$ be a family of priors on $\Lambda$. Then $\mathscr{F}$ is said to be conjugate for $\mathscr{K}$ if it is closed under sampling, that is, if the posterior distributions of $\lambda$ given the data $x$ belong to $\mathscr{F}$ for almost all $x$. In this paper, we set up a framework for the study of what we term the dual problem: for a given family of priors $\mathscr{F}$ (a subfamily of a general exponential family), find the class of sampling models $\mathscr{K}$ for which $\mathscr{F}$ is conjugate. In particular, we show that $\mathscr{K}$ must be a general exponential family dominated by some measure $Q$ on $(\mathscr{X}, B),$ where $B$ is the Borel field on $\mathscr{X}$. It is the class of such measures $Q$ that we investigate in this paper. We study its geometric features and general structure and apply the results to some familiar examples.


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Shaul K. Bar-Lev. Peter Enis. Gerard Letac. "Sampling Models which Admit a Given General Exponential Family as a Conjugate Family of Priors." Ann. Statist. 22 (3) 1555 - 1586, September, 1994.


Published: September, 1994
First available in Project Euclid: 11 April 2007

zbMATH: 0827.62002
MathSciNet: MR1311990
Digital Object Identifier: 10.1214/aos/1176325643

Primary: 62A15
Secondary: 60E99

Rights: Copyright © 1994 Institute of Mathematical Statistics


Vol.22 • No. 3 • September, 1994
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