用贝叶斯更新定义概率分布的类比关系，拓展了比例类比在概率领域的应用。
Proportional Analogies on Probability Distributions via Bayesian Updating
- 基于贝叶斯更新构建概率分布间的类比关系
- 在指数族分布上验证框架有效性，可扩展至任意分布
- 为概率推理提供新视角，适合概率建模与认知计算研究者
类比是形如‘A 对于 B 如同 C 对于 D’的四元关系。在各类类比推理形式中，比例类比通过一组公理刻画有效类比，具有重要理论意义。尽管比例类比在布尔、符号和实数域已广泛研究，其在概率分布上的扩展仍基本未被探索。本文提出一种基于贝叶斯更新的概率分布比例类比概念：两个分布相关当且仅当可通过一组合适观测诱导的贝叶斯更新相互转化。我们针对指数族中的典型分布进行分析，并讨论如何通过高斯混合近似将该框架推广至任意概率分布。
原文摘要 · Abstract (English)
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.
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