Bimonoidal Structure of Probability Monads

TL;DR

This paper develops a categorical framework for probability monads using bimonoidal structures to model joints, marginals, and independence, extending the theory beyond cartesian monoidal categories.

Contribution

It introduces bimonoidal structures for probability monads in non-cartesian monoidal categories, providing a unified approach to independence and probabilistic concepts.

Findings

Bimonoidal structures characterize joints and independence in categorical probability.

The Kantorovich monad on metric spaces is shown to be bimonoidal.

The framework generalizes probabilistic concepts beyond cartesian categories.

Abstract

We give a conceptual treatment of the notion of joints, marginals, and independence in the setting of categorical probability. This is achieved by endowing the usual probability monads (like the Giry monad) with a monoidal and an opmonoidal structure, mutually compatible (i.e. a bimonoidal structure). If the underlying monoidal category is cartesian monoidal, a bimonoidal structure is given uniquely by a commutative strength. However, if the underlying monoidal category is not cartesian monoidal, a strength is not enough to guarantee all the desired properties of joints and marginals. A bimonoidal structure is then the correct requirement for the more general case. We explain the theory and the operational interpretation, with the help of the graphical calculus for monoidal categories. We give a definition of stochastic independence based on the bimonoidal structure, compatible with the intuition and with other approaches in the literature for cartesian monoidal categories. We then show as an example that the Kantorovich monad on the category of complete metric spaces is a bimonoidal monad for a non-cartesian monoidal structure.