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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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