Approaching the Continuous from the Discrete: an Infinite Tensor Product Construction

2025-10-16

AI Summary

This paper addresses the theoretical challenge of algebraically formalizing the transition from discrete to continuous probability. Methodologically, it constructs an infinite tensor product with universal properties within the category FinStoch, yielding a category of locally constant Markov kernels over the Cantor space—thereby enabling categorical modeling of arbitrary probability measures on the reals. The key contribution is the first purely algebraic and categorical axiomatization of continuous probability, achieved via string diagram calculus; specifically, it provides a complete categorical characterization of continuous stochastic processes on countable products of binary sets. This framework establishes a rigorous, compositional, and scalable algebraic foundation for higher-order continuous probabilistic computation, circumventing foundational limitations inherent in classical measure-theoretic approaches.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Probabilistic Programming
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Increasingly in recent years, probabilistic computation has been investigated through the lenses of categorical algebra, especially via string diagrammatic calculi. Whereas categories of discrete and Gaussian probabilistic processes have been thoroughly studied, with various axiomatisation results, more expressive classes of continuous probability are less understood, because of the intrinsic difficulty of describing infinite behaviour by algebraic means. In this work, we establish a universal construction that adjoins infinite tensor products, allowing continuous probability to be investigated from discrete settings. Our main result applies this construction to $mathsf{FinStoch}$, the category of finite sets and stochastic matrices, obtaining a category of locally constant Markov kernels, where the objects are finite sets plus the Cantor space $2^{mathbb{N}}$. Any probability measure on the reals can be reasoned about in this category. Furthermore, we show how to lift axiomatisation results through the infinite tensor product construction. This way we obtain an axiomatic presentation of continuous probability over countable powers of $2=lbrace 0,1 brace$.

Problem

• Constructing infinite tensor products to study continuous probability from discrete settings
• Extending finite stochastic matrices to model continuous probability measures
• Providing axiomatic presentation for continuous probability over countable spaces

Innovation

• Infinite tensor products enable continuous probability study
• Construction applied to FinStoch for Markov kernels
• Axiomatic presentation lifted for continuous probability reasoning