Bundles of Probability Schemes

TL;DR

This paper introduces a categorical framework called bundles for finite probability theory, unifying concepts like conditional expectation, independence, and Markov chains.

Contribution

It develops a novel categorical approach that captures key probabilistic concepts and laws within a unified structure called bundles.

Findings

Provides a compact construction of conditional expectation.

Recovers laws of total expectation, variance, and covariance.

Models conditional independence and Markov chains using fiber products.

Abstract

We study finite probability theory through a category of finite probability schemes and probability-preserving maps, called \emph{bundles}. A bundle simultaneously records a quotient of a sample space, an algebra of random variables, and the family of conditional schemes over the quotient. The two natural linear functors associated with a bundle give a compact construction of conditional expectation and explain its projection properties. Within this framework we recover the laws of total expectation, variance, and covariance, the weak law of large numbers, and the variance decomposition behind simple linear regression. Fiber products then encode conditional independence and discrete-time Markov chains.