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Sampling models which admit a given general exponential family as a conjugate family of priors. (English) Zbl 0827.62002
Summary: Let \({\mathcal K}= \{K_\lambda\): \(\lambda\in \Lambda\}\) be a family of sampling distributions for the data \(x\) on a sample space \({\mathcal X}\) which is indexed by a parameter \(\lambda\in \Lambda\), and let \({\mathcal F}\) be a family of priors on \(\Lambda\). Then \({\mathcal F}\) is said to be conjugate for \({\mathcal K}\) if it is closed under sampling, that is, if the posterior distributions of \(\lambda\) given the data \(x\) belong to \({\mathcal F}\) for almost all \(x\).
We set up a framework for the study of what we term the dual problem: for a given family of priors \({\mathcal F}\) (a subfamily of a general exponential family), find the class of sampling models \({\mathcal K}\) for which \({\mathcal F}\) is conjugate. In particular, we show that \({\mathcal K}\) must be a general exponential family dominated by some measure \(Q\) on \(({\mathcal X}, {\mathcal B})\), where \({\mathcal B}\) is the Borel field on \({\mathcal X}\). It is the class of such measures \(Q\) that we investigate. We study its geometric features and general structure and apply the results to some familiar examples.

MSC:
62A01 Foundations and philosophical topics in statistics
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