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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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