On the evolution of the concept of probability as a mirror of the evolution of reason
Authors: Jean-Louis Le Mouël, Vincent Courtillot, Dominique Gibert, Vladimir Kossobokov, Jean-Baptiste Boulé, Pierpaolo Zuddas, Fernando Lopes, Païkan Marccagi, +1 more
Organizations: Acad´emie des Sciences, Institut de France, Paris, France · DeepField Sensing, France · Institute of Earthquake Prediction Theory and Mathematical Geophysics, Russian Academy of Sciences, Moscow, Russia · Accademia Nazionale delle Scienze detta dei XL, Roma, Italia · Sorbonne Universit´e, CNRS, METIS,UMR7619, Paris, France · Mus´eum National d’Histoire Naturelle, CNRS UMR7196, INSERM U1154, Paris, France · Laboratoire de G´eologie de l’ENS, UMR 8538, Paris, France
Abstract
Over the centuries, probability theory has grown from the calculus of games of chance into a central framework for reasoning under uncertainty. This article interprets that evolution not merely as a mathematical history, but as a transformation of rationality itself. From Pascal and Fermat's combinatorial symmetry to the inductive logic of Bayes and Laplace, from Poisson's statistics of events to Kolmogorov's axiomatic formalization, probability progressively incorporated uncertainty, time, and coherence into scientific judgment. This trajectory reaches a mature epistemological form in modern Bayesian inference, especially in Tarantola's view of probability as a logic of information, where prior knowledge and data are combined coherently. Yet this framework also exposes a limit: probability quantifies uncertainty about well-defined propositions, but does not by itself formalize the vagueness of the concepts used to describe them. The article therefore examines how rationality extends beyond probability. Fuzzy logic is presented as a rigorous language for graded meaning and qualitative judgment, while deep learning is analyzed as a distinct, powerful mode of prediction based on geometric interpolation and optimization rather than explicit inference. By situating probability, fuzzy logic, and deep learning in a common historical and epistemological perspective, the article clarifies their roles and limits. It argues that contemporary scientific rationality cannot be reduced to data-driven performance alone, but requires the explicit articulation of uncertainty, vagueness, and inference.
Although probabilistic statements are ubiquitous, foundational disagreements persist about their understanding, as exemplified by debates between Bayesians and frequentists; moreover, it is unclear when and why acting on them actually leads to desirable outcomes. Here, we argue that every probability is the output of a \emph{prediction method}, that is, it depends on both a particular way of constructing abstractions and a way of transforming them into predictions. Through this, we provide a unifying perspective on supposedly different kinds of probabilities and show that even supposedly objective ones are model-dependent. We demonstrate that when a finite calibration criterion is met, one can anticipate the distribution of utilities for a given policy and inform successful decision-making on finite sets of events. Based on the notion of prediction methods, inductive arguments, and the probability calculus, we explain the feasibility of the calibration criterion in many settings. Overall, we develop a coherent perspective on probabilities and their use, connecting key intuitions behind other interpretations along the way.
Analogies are quaternary relations of the form "A is to B as C is to D". Among the various formalizations of analogical reasoning, proportional analogies provide an important axiomatic framework by characterizing valid analogies through a set of postulates. While proportional analogies have been extensively studied over Boolean, symbolic, and real-valued domains, their extension to probability distributions remains largely unexplored. In this paper, we introduce a notion of proportional analogy for probability distributions based on Bayesian updating. Our approach builds upon the idea that two distributions are related whenever one can be transformed into the other through Bayesian updating induced by a suitable set of observations. We investigate this framework for several standard members of the exponential family and discuss how it naturally extends to arbitrary probability distributions through Gaussian mixture approximations.
Current evaluation of epistemic uncertainty relies on tasks such as out-ofdistribution detection and active learning. However, the Bayes-optimal decision strategies for these tasks do not coincide with the scores commonly used to quantify epistemic uncertainty. Building on the epistemic reject-option framework, we evaluate epistemic uncertainty using its ability to identify regret, the reducible error. Formulating selective prediction as a constrained optimization over coverage, expected risk, and regret, we prove the optimal selector is a thresholded convex combination of the ground-truth aleatoric and epistemic uncertainties. This theoretical unification exposes a weakness in recent uncertainty disentanglement literature: we demonstrate that standard correlation metrics between learned components do not necessarily predict their actual operational utility. We instead propose to evaluate the achievable risk, regret, coverage surface of the decomposition as a diagnostic for joint disentanglement and utility. Benchmarking standard methods on datasets with dense human annotations reveals that decision-theoretic rankings can disagree substantially with proxy-task rankings, including pairwise rank inversions between methods that are top-ranked on one criterion and bottom-ranked on other.
Jakub Paplhám, Willem Waegeman, Eyke Hüllermeier +1