From numerical proportions to analogical proportions between probabilities
2026-06-22 • Artificial Intelligence
Artificial Intelligence
AI summaryⓘ
The authors explore how analogical proportions, a way of relating four items as "a is to b as c is to d," can be applied in a probabilistic context. They extend previous work that dealt with analogies between fixed values or vectors to the case where each item is a probability or a probability distribution. The paper studies different mathematical ways to define such analogical proportions between probabilities and tests whether distributions linked to profiles that form analogical proportions also exhibit this property. This work aims to better understand how analogies work when uncertainty or variability is involved, such as with frequency distributions.
Analogical ProportionsProbability DistributionsArithmetic ProportionGeometric ProportionDiscrete AttributesClassificationNormalizationProfilesVectorsFrequency Distribution
Authors
Henri Prade, Gilles Richard
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.