乘性影响图中期望效用计算的符号代数
A symbolic algebra for the computation of expected utilities in multiplicative influence diagrams
摘要：
影响图为决策问题提供了一种简洁的图形表示。在文献中有几种算法可以快速计算它们的相关预期效用。然而，它们通常依赖于概率不确定性和效用值的全面量化。对于所有随机变量和决策空间都是有限和离散的问题，在这里，我们开发了一种符号化的方法来计算影响图的预期效用，不需要完整的数值表示。在这种方法中，预期效用对应于多项式族。在描述了它们的多项式结构之后，我们开发了一个有效的符号算法，用于通过图传播期望效用，并提供了一个使用计算机代数系统的算法的实现。然后，我们描述了许多标准的操作的影响图的多项式变换。我们还概括了这些图的决策分析框架，通过定义不对称的操作在预期的效用多项式。
英文摘要：
Influence diagrams provide a compact graphical representation of decision problems. Several algorithms for the quick computation of their associated expected utilities are available in the literature. However, often they rely on a full quantification of both probabilistic uncertainties and utility values. For problems where all random variables and decision spaces are finite and discrete, here we develop a symbolic way to calculate the expected utilities of influence diagrams that does not require a full numerical representation. Within this approach expected utilities correspond to families of polynomials. After characterizing their polynomial structure, we develop an efficient symbolic algorithm for the propagation of expected utilities through the diagram and provide an implementation of this algorithm using a computer algebra system. We then characterize many of the standard manipulations of influence diagrams as transformations of polynomials. We also generalize the decision analytic framework of these diagrams by defining asymmetries as operations over the expected utility polynomials.
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