From numerical proportions to analogical proportions between probabilities

• URL: http://arxiv.org/abs/2606.23029v1
• Date: Mon, 22 Jun 2026 08:39:28 GMT
• Title: From numerical proportions to analogical proportions between probabilities
• Abstract summary: Analogical proportions link four items a, b, c, d by a relation stating that a is to b as c is to d"<n>In this paper, we consider another representation setting, which is the probabilistic one.
• Score: 2.9649783577150832
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• Proportional Analogies on Probability Distributions via Bayesian Updating [13.360204757207262]
  We introduce a notion of proportional analogy for probability distributions based on Bayesian updating.<n>Our approach builds upon the idea that two distributions are related whenever one can be transformed into the other.<n>We discuss how it naturally extends to arbitrary probability distributions through Gaussian mixture approximations.
  arXiv  Detail & Related papers  (2026-08-12T07:02:43Z)
• Computational Identifiability [58.12019389497595]
  'Identical identifiability' assumes properties, infinite data, or other mathematically idealized conditions.<n>'Computational identifiability' is to instead define a finite computational search procedure for an empirical estimator.<n>We show how this framework allows us to answer fine-grained, practical identification questions.
  arXiv  Detail & Related papers  (2026-06-08T19:39:59Z)
• When Does Closeness in Distribution Imply Representational Similarity? An Identifiability Perspective [9.578534178372829]
  We prove that a small Kullback--Leibler divergence between the model distributions does not guarantee that the corresponding representations are similar.<n>We then define a distributional distance for which closeness implies representational similarity.<n>In synthetic experiments, we find that wider networks learn distributions which are closer with respect to our distance and have more similar representations.
  arXiv  Detail & Related papers  (2025-06-04T09:44:22Z)
• Some recent advances in reasoning based on analogical proportions [9.861775841965386]
  Analogical proportions play a key role in the formalization of analogical inference. The paper first discusses how to improve analogical inference in terms of accuracy and in terms of computational cost.
  arXiv  Detail & Related papers  (2022-12-22T14:10:14Z)
• Statistical Efficiency of Score Matching: The View from Isoperimetry [96.65637602827942]
  We show a tight connection between statistical efficiency of score matching and the isoperimetric properties of the distribution being estimated. We formalize these results both in the sample regime and in the finite regime.
  arXiv  Detail & Related papers  (2022-10-03T06:09:01Z)
• Proportoids [0.0]
  Analogical proportions are expressions of the form $a$ is to $b$ what $c$ is to $d$'' at the core of analogical reasoning. This paper contributes to the mathematical foundations of analogical proportions in the axiomatic tradition.
  arXiv  Detail & Related papers  (2022-08-31T00:04:34Z)
• Nonparametric Conditional Local Independence Testing [69.31200003384122]
  Conditional local independence is an independence relation among continuous time processes. No nonparametric test of conditional local independence has been available. We propose such a nonparametric test based on double machine learning.
  arXiv  Detail & Related papers  (2022-03-25T10:31:02Z)
• Sampling from Arbitrary Functions via PSD Models [55.41644538483948]
  We take a two-step approach by first modeling the probability distribution and then sampling from that model. We show that these models can approximate a large class of densities concisely using few evaluations, and present a simple algorithm to effectively sample from these models.
  arXiv  Detail & Related papers  (2021-10-20T12:25:22Z)
• Interaction Models and Generalized Score Matching for Compositional Data [9.797319790710713]
  We propose a class of exponential family models that accommodate general patterns of pairwise interaction while being supported on the probability simplex. Special cases include the family of Dirichlet distributions as well as Aitchison's additive logistic normal distributions. A high-dimensional analysis of our estimation methods shows that the simplex domain is handled as efficiently as previously studied full-dimensional domains.
  arXiv  Detail & Related papers  (2021-09-10T05:29:41Z)
• Probabilistic Generating Circuits [50.98473654244851]
  We propose probabilistic generating circuits (PGCs) for their efficient representation. PGCs are not just a theoretical framework that unifies vastly different existing models, but also show huge potential in modeling realistic data. We exhibit a simple class of PGCs that are not trivially subsumed by simple combinations of PCs and DPPs, and obtain competitive performance on a suite of density estimation benchmarks.
  arXiv  Detail & Related papers  (2021-02-19T07:06:53Z)
• A Complete Characterization of Projectivity for Statistical Relational Models [20.833623839057097]
  We introduce a class of directed latent graphical variable models that precisely correspond to the class of projective relational models. We also obtain a characterization for when a given distribution over size-$k$ structures is the statistical frequency distribution of size-$k$ sub-structures in much larger size-$n$ structures.
  arXiv  Detail & Related papers  (2020-04-23T05:58:27Z)
• Profile Entropy: A Fundamental Measure for the Learnability and Compressibility of Discrete Distributions [63.60499266361255]
  We show that for samples of discrete distributions, profile entropy is a fundamental measure unifying the concepts of estimation, inference, and compression. Specifically, profile entropy a) determines the speed of estimating the distribution relative to the best natural estimator; b) characterizes the rate of inferring all symmetric properties compared with the best estimator over any label-invariant distribution collection; c) serves as the limit of profile compression.
  arXiv  Detail & Related papers  (2020-02-26T17:49:04Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.