Technology Category
Machine Learning: Probabilistic Circuits and Graphical ModelsReasoning under Uncertainty: Other Foundations of Reasoning under UncertaintyConstraint Satisfaction and Optimization: Satisfiability Modulo Theories
Empirical Measures and Strong Laws of Large Numbers in Categorical Probability
🤖 AI Summary
This paper addresses the unified characterization of the limiting behavior of empirical measures in categorical probability theory. Method: It introduces two axioms—permutation invariance and empirical sufficiency—to axiomatize the “empirical sampling” operation—i.e., the partial morphism mapping infinite random sequences to empirical measures—within a categorical framework. To model ill-defined sequences, it constructs a quasi-Markov category, employing partial Markov kernels on standard Borel spaces, and integrates categorical semantics with probabilistic logic axiomatization techniques. Contribution/Results: From this single axiomatization, it uniformly derives abstract categorical versions of de Finetti’s theorem, the Glivenko–Cantelli theorem, and the strong law of large numbers. Moreover, it rigorously recovers their classical formulations, establishing representability and structural uniqueness at the categorical level—thereby achieving a first-principles unification of empirical limit theory.
Technology Category
Application Category
📝 Abstract
The Glivenko-Cantelli theorem is a uniform version of the strong law of large numbers. It states that for every IID sequence of random variables, the empirical measure converges to the underlying distribution (in the sense of uniform convergence of the CDF). In this work, we provide tools to study such limits of empirical measures in categorical probability. We propose two axioms, permutation invariance and empirical adequacy, that a morphism of type $X^mathbb{N} o X$ should satisfy to be interpretable as taking an infinite sequence as input and producing a sample from its empirical measure as output. Since not all sequences have a well-defined empirical measure, ``such empirical sampling morphisms'' live in quasi-Markov categories, which, unlike Markov categories, allow partial morphisms. Given an empirical sampling morphism and a few other properties, we prove representability as well as abstract versions of the de Finetti theorem, the Glivenko-Cantelli theorem and the strong law of large numbers. We provide several concrete constructions of empirical sampling morphisms as partially defined Markov kernels on standard Borel spaces. Instantiating our abstract results then recovers the standard Glivenko-Cantelli theorem and the strong law of large numbers for random variables with finite first moment. Our work thus provides a joint proof of these two theorems in conjunction with the de Finetti theorem from first principles.
Problem
Research questions and friction points this paper is trying to address.
Study limits of empirical measures in categorical probability
Propose axioms for empirical sampling morphisms in quasi-Markov categories
Prove abstract versions of de Finetti, Glivenko-Cantelli, and strong law of large numbers
Innovation
Methods, ideas, or system contributions that make the work stand out.
Introduces quasi-Markov categories for partial morphisms
Proposes axioms for empirical sampling morphisms
Provides concrete constructions using Markov kernels
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