From Moore-Penrose to Markov via Gauss

TL;DR

This paper introduces the Gauss construction on Moore-Penrose dagger additive categories to generate Markov categories with conditionals, extending Gaussian probability theory to complex and quaternionic matrices.

Contribution

It formalizes a new method to construct Markov categories with conditionals from Moore-Penrose categories, broadening the scope of Gaussian probability models.

Findings

Recovers Gaussian probability theory via the Gauss construction on real matrices.

Extends the framework to complex and quaternionic matrices, creating new Markov categories.

Characterizes all possible conditionals within the Gauss construction.

Abstract

Markov categories are the central framework for categorical probability theory. Many important concepts from probability theory can be formalized in terms of Markov categories. In particular, conditional probability distributions and Bayes' theorem are captured via the notion of conditionals in a Markov category. Gaussian probability theory gives an example of a Markov category with conditionals, where the conditionals can be computed using the Moore-Penrose inverse. In this paper, we introduce the Gauss construction on a Moore-Penrose dagger additive category, producing a Markov category with conditionals. Applying the Gauss construction to the category of real matrices recaptures the Gaussian probability theory example, while applying it to the category of complex (resp. quaternionic) matrices gives us new Markov categories of proper complex (resp. quaternionic) Gaussian conditional distributions. Moreover, we also characterize all possible conditionals in the Gauss construction.