From numerical proportions to analogical proportions between probabilities

2026-06-22

AI Summary

This study addresses the formalization of analogical proportions in the space of probability distributions, specifically tackling the challenge of defining and applying such proportions between discrete attribute distributions. It systematically extends analogical proportions from scalar values to normalized probability distributions for the first time, proposing a modeling approach that integrates both arithmetic and geometric proportionality to capture analogical structures among distributions. Theoretical analysis and empirical experiments demonstrate that when four feature vectors satisfy analogical proportions component-wise, their corresponding discrete probability distributions tend to preserve analogous relationships. The proposed framework effectively supports analogical reasoning and classification tasks, thereby validating the feasibility and practical utility of probabilistic analogies in predictive settings.

Abstract

Analogical proportions link four items a, b, c, d by a relation stating that ``a is to b as c is to d", a, b, c, d being the formal representation of real world entities, ranging from simple numerical values to more complex structures such as profiles. Accordingly, $a, b, c, d$ could be atomic values like Boolean, nominal or numerical values, more generally vectors of such values, or even families of items represented by logical formulas. In this paper, we consider another representation setting, which is the probabilistic one. Precisely, the article proposes a study of {analogical} proportions between probabilities, whether they are simply between probability values, or between distributions (which requires the preservation of their normalization). More particularly, we study the properties of definitions based on arithmetic proportion, or on a combination of the former with geometric proportion, while other options are also discussed. Previous works have shown that when four profiles a, b, c, d, represented as vectors, form analogical proportions componentwise, it is likely that their classes form an analogical proportion also. This is the basis of an analogical proportion-based classification method that can produce accurate predictions. Similarly, in this paper, each profile is associated with a distribution describing the frequencies of the possible values of a discrete attribute of interest. We then discuss and experimentally investigate if the distributions associated to four profiles forming an analogical proportion themselves also form an analogical proportion.

Problem

• analogical proportions
• probabilities
• probability distributions
• classification
• discrete attributes

Innovation

• analogical proportions
• probabilistic representation
• probability distributions
• arithmetic proportion
• classification