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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.

62A01 Foundations and philosophical topics in statistics
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