To model discrete-time finite-state uncertain processes, we argue for the use of a global belief model in the form of an upper expectation that is the most conservative one under a set of basic axioms. Our motivation for these axioms, which describe how local and global belief models should be related, is based on two possible interpretations for an upper expectation: a behavioural one similar to Walley's, and an interpretation in terms of upper envelopes of linear expectations. We show that the most conservative upper expectation satisfying our axioms, that is, our model of choice, coincides with a particular version of the game-theoretic upper expectation introduced by Shafer and Vovk. This has two important implications: it guarantees that there is a unique most conservative global belief model satisfying our axioms; and it shows that Shafer and Vovk's model can be given an axiomatic characterisation and thereby provides an alternative motivation for adopting this model, even outside their game-theoretic framework. Finally, we relate our model to the upper expectation resulting from a traditional measure-theoretic approach. We show that this measure-theoretic upper expectation also satisfies the proposed axioms, which implies that it is dominated by our model or, equivalently, the game-theoretic model. Moreover, if all local models are precise, all three models coincide.

中文翻译：

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作为离散时间有限状态不确定过程的全局信念模型的特定上限期望

为了对离散时间有限状态不确定过程进行建模，我们主张以上期望的形式使用全局置信模型，该模型是一组基本公理下最保守的模型。我们对这些描述局部和全局信念模型应该如何关联的公理的动机基于对上限期望的两种可能解释：类似于沃利的行为解释，以及线性期望上限的解释。我们表明，满足我们公理的最保守的上限期望，即我们的选择模型，与 Shafer 和 Vovk 引入的博弈论上限期望的特定版本一致。这有两个重要的含义：它保证有一个独特的最保守的全局信念模型满足我们的公理；并且它表明 Shafer 和 Vovk 的模型可以被赋予公理化特征，从而为采用该模型提供了另一种动机，即使在他们的博弈论框架之外。最后，我们将我们的模型与传统测量理论方法产生的上期望联系起来。我们表明，这个度量理论的上限期望也满足提出的公理，这意味着它由我们的模型或等效的博弈论模型主导。此外，如果所有局部模型都是精确的，则所有三个模型都是一致的。我们将我们的模型与传统测度理论方法产生的上期望值联系起来。我们表明，这个度量理论的上限期望也满足提出的公理，这意味着它由我们的模型或等效的博弈论模型主导。此外，如果所有局部模型都是精确的，则所有三个模型都是一致的。我们将我们的模型与传统测量理论方法产生的上期望值联系起来。我们表明，这个度量理论的上限期望也满足提出的公理，这意味着它由我们的模型或等效的博弈论模型主导。此外，如果所有局部模型都是精确的，则所有三个模型都是一致的。