Bundles of Probability Schemes

Abstract

We study finite probability through a category whose objects are finite probability schemes and whose morphisms are probability-preserving maps, called bundles. A bundle simultaneously records a quotient of a sample space, the corresponding algebra of random variables, and the conditional schemes on its fibers. The pullback, transfer, and fiberwise averaging maps associated with a bundle give a functorial construction of conditional expectation and its projection properties. This recovers serveral fundamental results in elementary probability. Fiber products of bundles encode conditional independence, and iterated fiber products describe finite Markov chains from their adjacent-pair distribution schemes. Finally, the transfer map is identified with a relative trace, so the projection formula and base-change identity become instances of Frobenius reciprocity and Beck--Chevalley condition.