We examine general decision problems with loss functions that are bounded below. We allow the loss function to assume the value \(\infty \). No other assumptions are made about the action space, the types of data available, the types of non-randomized decision rules allowed, or the parameter space. By allowing prior distributions and the randomizations in randomized rules to be finitely-additive, we prove very general complete class and minimax theorems. Specifically, under the sole assumption that the loss function is bounded below, we show that every decision problem has a minimal complete class and all admissible rules are Bayes rules. We also show that every decision problem has a minimax rule and a least-favorable distribution and that every minimax rule is Bayes with respect to the least-favorable distribution. Some special care is required to deal properly with infinite-valued risk functions and integrals taking infinite values.

中文翻译：

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有限可加性可以为决策理论增加什么

我们研究具有以下限制的损失函数的一般决策问题。我们允许损失函数采用值\（\ infty \）。关于动作空间，可用数据的类型，允许的非随机决策规则的类型或参数空间，没有其他假设。通过允许先验分布和随机规则中的随机化是有限可加的，我们证明了非常通用的完全类和极小极大定理。具体而言，在唯一的假设下，损失函数在下面受限制的情况下，我们表明，每个决策问题都有一个最小的完整类，并且所有可接受的规则都是贝叶斯规则。我们还表明，每个决策问题都有一个极小极大值规则和一个最不利的分布，并且相对于极不有利的分布，每个极小极大规则都是贝叶斯。