基于贝叶斯更新的概率分布比例类比
Proportional Analogies on Probability Distributions via Bayesian Updating
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中文总结 AI 辅助
本文针对概率分布的比例类比问题，基于贝叶斯更新提出相关概念，研究指数族分布并通过高斯混合近似扩展至任意概率分布，填补了该方向的研究空白。
中文摘要 AI 辅助
类比是形如“A之于B如同C之于D”的四元关系。在类比推理的多种形式化方法中，比例类比通过一组公设刻画有效类比，提供了重要的公理框架。尽管比例类比已在布尔、符号和实值域中得到广泛研究，但其向概率分布的扩展仍基本未被探索。本文中，我们基于贝叶斯更新引入概率分布的比例类比概念，该方法基于如下思想：当一个分布可通过合适观测集诱导的贝叶斯更新转换为另一个分布时，二者存在关联。我们针对指数族的若干标准成员研究该框架，并讨论其如何通过高斯混合近似自然扩展至任意概率分布。
英文摘要
Analogies are quaternary relations of the form "A is to B as C is to D". Among the various formalizations of analogical reasoning, proportional analogies provide an important axiomatic framework by characterizing valid analogies through a set of postulates. While proportional analogies have been extensively studied over Boolean, symbolic, and real-valued domains, their extension to probability distributions remains largely unexplored. In this paper, we introduce a notion of proportional analogy for probability distributions based on Bayesian updating. Our approach builds upon the idea that two distributions are related whenever one can be transformed into the other through Bayesian updating induced by a suitable set of observations. We investigate this framework for several standard members of the exponential family and discuss how it naturally extends to arbitrary probability distributions through Gaussian mixture approximations.
