ABSTRACT The main contribution of this paper is a new definition of expected value of belief functions in the Dempster–Shafer (D–S) theory of evidence. Our definition shares many of the properties of the expectation operator in probability theory. Also, for Bayesian belief functions, our definition provides the same expected value as the probabilistic expectation operator. A traditional method of computing expected of real-valued functions is to first transform a D–S belief function to a corresponding probability mass function, and then use the expectation operator for probability mass functions. Transforming a belief function to a probability function involves loss of information. Our expectation operator works directly with D–S belief functions. Another definition is using Choquet integration, which assumes belief functions are credal sets, i.e. convex sets of probability mass functions. Credal sets semantics are incompatible with Dempster's combination rule, the center-piece of the D–S theory. In general, our definition provides different expected values than, e.g. if we use probabilistic expectation using the pignistic transform or the plausibility transform of a belief function. Using our definition of expectation, we provide new definitions of variance, covariance, correlation, and other higher moments and describe their properties.

中文翻译：

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Dempster-Shafer 理论中信念函数的期望算子

摘要 本文的主要贡献是在 Dempster-Shafer (D-S) 证据理论中对信念函数的期望值进行了新的定义。我们的定义与概率论中期望算子的许多属性相同。此外，对于贝叶斯置信函数，我们的定义提供了与概率期望算子相同的期望值。计算实值函数期望值的传统方法是先将 D-S 置信函数转换为对应的概率质量函数，然后对概率质量函数使用期望算子。将置信函数转换为概率函数涉及信息丢失。我们的期望算子直接与 D-S 信念函数一起工作。另一个定义是使用 Choquet 积分，它假设信念函数是信任集，即 概率质量函数的凸集。Credal 集语义与 Dempster 的组合规则（D-S 理论的核心部分）不兼容。一般来说，我们的定义提供了不同的期望值，例如，如果我们使用概率期望，使用 pignistic 变换或置信函数的似真变换。使用我们对期望的定义，我们提供了方差、协方差、相关性和其他更高矩的新定义，并描述了它们的属性。