## Fully coherent inference.(English)Zbl 0737.62001

Given a sample space $$(X,{\mathcal G})$$, the parameter space $$(\Theta,{\mathcal F})$$, and the sampling model $$p( |\theta)$$, a statistical inference $$q$$ is a kernel on $${\mathcal F}\times X$$ which is used to construct a system of conditional probabilities in Rényi’s sense. The prior $$\mu$$ on $$\Theta$$ is minimally compatible with $$q$$ if $$\mu(A)>0$$ for every $$A\subseteq K$$, $$\mu(K)<\infty$$, for which there exists $$B\in{\mathcal G}$$ such that $$p(B\mid\theta)>0$$ for $$\theta\in A$$, and $$q(A\mid x)>0$$ for all $$x\in B$$. For $${\mathcal K}\subseteq{\mathcal F}$$ the inference $$q$$ is called fully $${\mathcal K}$$-expectation consistent with $$p$$ if for every $$K\in{\mathcal K}$$ there exists a probability concentrated on $$K$$ such that an expected nonnegative loss according to $$p$$ given $$\theta$$ is necessarily 0 a.s. on $$K$$ provided the expected loss according to $$q$$ given $$x$$ is 0 for every $$x\in X$$.
The first theorem states that for $$q$$ being $${\mathcal K}$$-posterior of $$p$$ and $$\mu$$ minimally $${\mathcal K}$$-compatible with $$q$$ it follows that $$q$$ is fully $${\mathcal K}$$-expectation consistent. The converse direction is shown to hold under some regularity conditions on the spaces $$X$$, $$\Theta$$ and a continuity condition on the sampling distribution.
Reviewer: F.Liese (Rostock)

### MSC:

 62A01 Foundations and philosophical topics in statistics 60A05 Axioms; other general questions in probability
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