通过贝叶斯更新实现概率分布上的比例类比
文章背景与核心概要
类比推理作为人工智能和认知科学的核心概念，长期以来在布尔逻辑、符号逻辑及实数域中得到了广泛研究。然而，如何将“A之于B犹如C之于D”的比例类比（Proportional Analogies）形式化并扩展到概率分布领域，此前一直处于未被充分探索的空白状态。
本文作者 Pierre-Alexandre Murena 提出了一种基于贝叶斯更新的概率分布比例类比新范式。该方法的核心思想是：当一个分布可以通过由一组适当观测诱导的贝叶斯更新转化为另一个分布时，这两个分布便建立起了类比关系。研究不仅在标准指数族的多个成员中验证了这一理论框架，还探讨了如何通过高斯混合近似将该方法自然扩展到任意概率分布。
Proportional Analogies on Probability Distributions via Bayesian Updating
arXiv: 2608.11724
Subject: Artificial Intelligence (cs.AI)
Author: Pierre-Alexandre Murena
Date: August 12, 2026
arXiv: 2608.11724
Subject: Artificial Intelligence (cs.AI)
Author: Pierre-Alexandre Murena
Date: August 12, 2026
Summary
This paper explores the formalization of proportional analogies—relations structured as "A is to B as C is to D"—within the domain of probability distributions. While such analogies are well-established in Boolean and symbolic logic, their application to probabilistic frameworks has remained largely uncharted. The author proposes a novel approach using Bayesian updating, where the relationship between two distributions is defined by the transformation induced by a specific set of observations. The study validates this framework across standard exponential family members and demonstrates its scalability to arbitrary distributions via Gaussian mixture approximations.
This paper explores the formalization of proportional analogies—relations structured as "A is to B as C is to D"—within the domain of probability distributions. While such analogies are well-established in Boolean and symbolic logic, their application to probabilistic frameworks has remained largely uncharted. The author proposes a novel approach using Bayesian updating, where the relationship between two distributions is defined by the transformation induced by a specific set of observations. The study validates this framework across standard exponential family members and demonstrates its scalability to arbitrary distributions via Gaussian mixture approximations.
Abstract
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.
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.
Access & Resources
- PDF: View Paper
- HTML: Experimental Version
- TeX Source: arXiv Source
- License: Creative Commons Attribution 4.0
- PDF: View Paper
- HTML: Experimental Version
- TeX Source: arXiv Source
- License: Creative Commons Attribution 4.0
Metadata
| Field | Details |
|---|---|
| Primary Subject | Artificial Intelligence (cs.AI) |
| DOI | 10.48550/arXiv.2608.11724 |
| Submission History | [v1] Wed, 12 Aug 2026 |
Field Details Primary Subject Artificial Intelligence (cs.AI) DOI 10.48550/arXiv.2608.11724 Submission History [v1] Wed, 12 Aug 2026