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**Coherent and continuous inference.**
*(English)*
Zbl 0555.62008

Suppose X and \(\Theta\) are the sample space of some experiment and the set of possible states of nature, respectively. Suppose \(p_{\theta}\) is the distribution of X given \(\theta\) and \(q_ x\) is a distribution on \(\Theta\) given x. If a ”bookie” uses \(q_ x\) to construct a conditional odds function to post odds on subsets of \(\Theta\) after observing X, then q is called coherent if it is impossible for a gambler to devise a betting scheme based on q consisting of placing a finite number of bets on subsets of \(\Theta\) after observing X and which has expected payoff bounded above 0 for all \(\theta\).

D. Heath and the second author [Ann. Stat. 6, 333-345 (1978; Zbl 0385.62005)] showed that coherent references are posterior distributions of proper, finitely additive priors on \(\Theta\). Here, the authors show the priors may be required to be countably additive, provided \(\Theta\) and X are separable metric spaces at least one of which is compact, and all \(p_{\theta}\) and \(q_ x\) are countably additive and weakly continuous. They include several informative examples and relate their work to that of several other authors. For example, they show coherence is equivalent to the denial of strong inconsistency, as defined in M. Stone, J. Am. Stat. Assoc. 71, 114-125 (1976; Zbl 0335.62026).

D. Heath and the second author [Ann. Stat. 6, 333-345 (1978; Zbl 0385.62005)] showed that coherent references are posterior distributions of proper, finitely additive priors on \(\Theta\). Here, the authors show the priors may be required to be countably additive, provided \(\Theta\) and X are separable metric spaces at least one of which is compact, and all \(p_{\theta}\) and \(q_ x\) are countably additive and weakly continuous. They include several informative examples and relate their work to that of several other authors. For example, they show coherence is equivalent to the denial of strong inconsistency, as defined in M. Stone, J. Am. Stat. Assoc. 71, 114-125 (1976; Zbl 0335.62026).

Reviewer: J.W.Green

### MSC:

62A01 | Foundations and philosophical topics in statistics |

60A05 | Axioms; other general questions in probability |