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This paper introduces a novel framework for understanding proportional analogies within the context of probability distributions, leveraging Bayesian updating as a foundational mechanism. By establishing that two distributions can be analogically related if one can be transformed into the other through Bayesian observations, the authors extend the concept of analogical reasoning beyond traditional domains. The key result demonstrates that this framework can be effectively applied to standard exponential family distributions and generalized to arbitrary distributions using Gaussian mixture approximations, thereby enriching the theoretical landscape of analogical reasoning in probabilistic contexts.
Transforming probability distributions through Bayesian updating reveals a new dimension of analogical reasoning that could redefine how we understand relationships in probabilistic models.
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.
