FROM NUMERICAL PROPORTIONS TO ANALOGICAL PROPORTIONS BETWEEN PROBABILITIES

This paper explores the intersection of analogical reasoning and probability theory. Specifically, it investigates how the concept of an analogical proportion—a relation stating that "a is to b as c is to d"—can be formally extended from simple numerical or categorical values to probability values and probability distributions. The authors aim to determine if four items forming an analogical proportion also result in their associated probability distributions forming an analogical proportion, thereby supporting analogical inference in classification tasks.

The authors build upon existing frameworks for analogical proportions, which are traditionally defined by postulates such as reflexivity, symmetry, and stability under central permutation. They extend these to the probabilistic domain by examining:

• Probability Values: Defining proportions between individual probabilities.
• Probability Distributions: Defining proportions between distributions, ensuring the preservation of normalization. The authors propose two primary definitions: one based on arithmetic proportions (which preserves the total variation distance) and one combining arithmetic and geometric proportions (which is consistent with the preservation of the Kullback-Leibler divergence).
• Empirical Validation: The authors test the hypothesis that if four profiles (represented as vectors) form an analogical proportion, their associated probability distributions will likely follow suit, validating this on large datasets.

The study demonstrates that analogical proportions can be rigorously defined for probabilities. By showing that these proportions can be maintained across distributions, the authors provide a formal justification for using analogical reasoning in probabilistic settings. This is significant because it bridges the gap between symbolic analogical inference and Bayesian-style probabilistic reasoning, offering a new mechanism for classification where the "analogical jump" can be applied to frequency distributions rather than just raw attribute values.