Categorical Probability Theory

TL;DR

This paper offers a categorical perspective on probability measures, demonstrating their structure as a submonad related to the Giry monad, and explores the integral operator within this framework.

Contribution

It introduces a categorical characterization of probability measures as a submonad of a double dualization monad, linking them to convex spaces and the Giry monad.

Findings

Probability measures form a submonad of a double dualization monad.

The submonad is isomorphic to the Giry monad.

A theorem on the integral operator's behavior within this categorical framework.

Abstract

We present a categorical viewpoint of probability measures by showing that a probability measure can be viewed as a weakly averaging affine measurable functional taking values in the unit interval which preserves limits. The probability measures on a space are the elements of a submonad of a double dualization monad on the category of measurable spaces into the unit interval, and this monad is naturally isomorphic to the Giry monad. We show this submonad is the codensity monad of a functor from the category of convex spaces to the category of measurable spaces. A theorem proving the integral operator acting on the space of measurable functions and the space of probability measures on the domain space of those functions is given using the strong monad structure of the Giry monad.