Picturing classical and quantum Bayesian inference

TL;DR

This paper develops a unified graphical framework for classical and quantum Bayesian inference using category theory, enabling a clearer understanding of inference structures and their quantum generalizations.

Contribution

It introduces a general diagrammatic approach to Bayesian inference that encompasses both classical and quantum cases, simplifying traditional representations and extending to quantum-like calculi.

Findings

Classical Bayesian inference characterized by a specific graphical property.

A noncommutative Frobenius structure models quantum Bayesian inference.

Framework connects to Bayesian networks and can be constructed in any dagger compact category.

Abstract

We introduce a graphical framework for Bayesian inference that is sufficiently general to accommodate not just the standard case but also recent proposals for a theory of quantum Bayesian inference wherein one considers density operators rather than probability distributions as representative of degrees of belief. The diagrammatic framework is stated in the graphical language of symmetric monoidal categories and of compact structures and Frobenius structures therein, in which Bayesian inversion boils down to transposition with respect to an appropriate compact structure. We characterize classical Bayesian inference in terms of a graphical property and demonstrate that our approach eliminates some purely conventional elements that appear in common representations thereof, such as whether degrees of belief are represented by probabilities or entropic quantities. We also introduce a quantum-like calculus wherein the Frobenius structure is noncommutative and show that it can accommodate Leifer's calculus of `conditional density operators'. The notion of conditional independence is also generalized to our graphical setting and we make some preliminary connections to the theory of Bayesian networks. Finally, we demonstrate how to construct a graphical Bayesian calculus within any dagger compact category.