# From numerical proportions to analogical proportions between probabilities

arXiv: https://arxiv.org/abs/2606.23029 · HTML: https://mlsanity.com/paper/2606.23029/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 7 of its 50 references that are in the map.
- First posted on arXiv 22 Jun 2026.

## Builds on (most cited first, 7 of 7)

- [Emergent Analogical Reasoning in Large Language Models](https://mlsanity.com/paper/2212.09196/) (2022)
- [On Wasserstein Two Sample Testing and Related Families of Nonparametric Tests](https://mlsanity.com/paper/1509.02237/) (2015)
- [A Countrywide Traffic Accident Dataset](https://mlsanity.com/paper/1906.05409/) (2019)
- [Analogical Dissimilarity: Definition, Algorithms and Two Experiments in Machine Learning](https://mlsanity.com/paper/1401.3427/) (2014)
- [A Description Logic for Analogical Reasoning](https://mlsanity.com/paper/2105.04620/) (2021)
- [Generalizing Analogical Inference from Boolean to Continuous Domains](https://mlsanity.com/paper/2511.10416/) (2025)
- [Frank's triangular norms in Piaget's logical proportions](https://mlsanity.com/paper/2408.03795/) (2024)

## Abstract (arXiv)

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.