Bayesian Updating Defines Proportional Analogies for Distributions

Analogical reasoning, often expressed as "A is to B as C is to D," is a fundamental cognitive process. While proportional analogies have been formally studied across various domains like Boolean logic, symbolic systems, and real numbers, their application to probability distributions has remained largely unexplored. This gap limits the ability of AI systems to reason analogically about uncertain information. This research proposes a new framework for defining proportional analogies specifically for probability distributions. The core idea is that two distributions are considered analogically related if one can be transformed into the other through a process of Bayesian updating, driven by a suitable set of observations. This approach leverages the well-established principles of Bayesian inference to establish relationships between probabilistic models. The paper investigates this framework for standard members of the exponential family of distributions and discusses its natural extension to arbitrary probability distributions using Gaussian mixture approximations. This work provides a foundational axiomatic framework for analogical reasoning in probabilistic domains, opening new avenues for AI systems to understand and generate analogies involving uncertainty.