On extended admissible procedures and their nonstandard Bayes risk
利用数理逻辑和非标准分析，证明在任意统计决策问题中，有限风险函数的决策程序是扩展可容许的当且仅当其贝叶斯风险为无穷小，无需正则条件。
For finite parameter spaces, among decision procedures with finite risk functions, a decision procedure is extended admissible if and only if it is Bayes. Various relaxations of this classical equivalence have been established for infinite parameter spaces, but these extensions are each subject to technical conditions that limit their applicability, especially to modern (semi and nonparametric) statistical problems. Using results in mathematical logic and nonstandard analysis, we extend this equivalence to arbitrary statistical decision problems: informally, we show that, among decision procedures with finite risk functions, a decision procedure is extended admissible if and only if it has infinitesimal excess Bayes risk. In contrast to existing results, our equivalence holds in complete generality, that is, without regularity conditions or restrictions on the model or loss function. We also derive a nonstandard analogue of Blyth’s method that yields sufficient conditions for admissibility, and apply the nonstandard theory to derive a purely standard theorem: when risk functions are continuous on a compact Hausdorff parameter space, a procedure is extended admissible if and only if it is Bayes.