Lane, David A.; Sudderth, William D. Coherent and continuous inference. (English) Zbl 0563.62003 Ann. Stat. 11, 114-120 (1983). Let \(\Theta\) and X denote the sets of possible states of nature and possible observations, respectively. A sampling model p assigns to each \(\theta\in \Theta\) an element \(p_{\theta}\) of the set P(X) of finitely additive probability measures defined on all subsets of X. An inference q assigns to \(x\in X\) an element \(q_ x\in P(\Theta)\). Let B(\(\Theta)\) and B(X) denote \(\sigma\)-algebras of subsets of \(\Theta\) and X, respectively. Two inferences p and q are said to be consistent if there exists \(r\in P(\Theta \times X)\) such that \[ r(\phi)=\int p_{\theta}(\phi_{\theta})\pi (d\theta)=\int q_ x(\phi^ x)m(dx) \] for every bounded \(B=B(\Theta)\otimes B(X)\)-measurable function \(\phi\) : \(\Theta\) \(\times X\to {\mathbb{R}}\), where \(\pi\) and m are the marginal distributions of r. For a given p, the inference q is coherent [for definition see the paper of D. C. Heath and the second author, ibid. 6, 333-345 (1978; Zbl 0385.62005)] if and only if p and q are consistent; p and q are strongly inconsistent in the sense of M. Stone [J. Am. Stat. Assoc. 71, 114-125 (1976; Zbl 0335.62026)] if and only if p and q are not consistent. Finally, \(\Theta\) and X are assumed to be separable metric spaces with their Borel sets; furthermore, p and q are continuous mappings to M(X), the set of countably additive probability measures on X with the weak topology, and to M(\(\Theta)\), respectively. Result: If either \(\Theta\) or X is compact, then q is coherent for p if and only if q is the posterior distribution of a proper, countably additive (!) prior distribution. Cited in 1 ReviewCited in 5 Documents MSC: 62A01 Foundations and philosophical topics in statistics Keywords:coherent and continuous inference; finitely additive probability measures; separable metric spaces; countably additive probability measures; weak topology; posterior distribution Citations:Zbl 0385.62005; Zbl 0335.62026 × Cite Format Result Cite Review PDF Full Text: DOI