Empirical Measures and Strong Laws of Large Numbers in Categorical Probability (2503.21576v1)

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 $X^\mathbb{N} \to X$ 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.

Analysis of Empirical Measures and Laws of Large Numbers in Categorical Probability

The paper "Empirical Measures and Strong Laws of Large Numbers in Categorical Probability" presents an exploration of empirical measures and associated strong laws of large numbers within the framework of categorical probability. This work aims to integrate these ideas into an advanced probabilistic structure, leveraging categorical formulations to offer precision and abstraction.

The primary focus of the paper is the formulation and examination of empirical measures through the lens of categorical probability, emphasizing the structured and abstract redeployment of these concepts. The authors introduce two axioms: permutation invariance and empirical adequacy, which are central to interpreting a morphism as a means of derivation from an infinite sequence to a sample via its empirical measure. These axioms facilitate a categorical structure that encapsulates empirical sampling, making the notion applicable in quasi-Markov categories that can handle partial morphisms.

Crucially, the research recovers and generalizes classical probabilistic results such as the Glivenko-Cantelli theorem, the de Finetti theorem, and the strong law of large numbers. These are shown within a categorical context, providing a synthetic validation that not only reproduces known results but also extends them into more abstract categorical territories, demonstrating the potential of using category theory to unify and broaden probabilistic theorems.

The authors offer a robust construction of empirical sampling morphisms within a standard Borel space. They ensure that these morphisms satisfy the prerequisites for being interpreted as empirical measures, extending their applicability beyond traditional spaces to all standard Borel spaces by constructing appropriate categorical analogs. This is achieved by identifying precisely which sequences have well-defined empirical measures and formulating them as partial Markov kernels in a quasi-Markov category framework.

Numerically, the paper is underlined by precise control of convergence and bounds related to empirical averages and measures, featuring universal conditions under which empirical processes exhibit stable convergence behaviors. One such result includes demonstrating that the strong laws of large numbers can be conceptually and quantitatively anchored in the proposed categorical framework, ensuring empirical averages converge under IID assumptions subject to finite moments.

Implications of this paper span both theoretical advancements in categorical probability and practical implementations in artificial intelligence, where probabilistic reasoning is paramount. Looking forward, the developed framework sets a solid foundation for exploring other stochastic processes categorically, potentially affecting how concepts like ergodicity and causal influence are interpreted in future AI systems.

In conclusion, the paper successfully integrates empirical measures and strong laws of large numbers into the categorical probability field, pushing the boundaries of traditional probability theory through a sophisticated and formal abstraction. The implications resonate with both immediate theoretical interest and long-term practical percolation into applications that require robust probabilistic inference mechanisms.