A Category for unifying Gaussian Probability and Nondeterminism

TL;DR

This paper introduces a new categorical framework that unifies Gaussian probability distributions with relational nondeterminism, enabling rigorous modeling of complex systems in statistics, engineering, and control theory.

Contribution

It presents categories of extended Gaussian maps and relations that combine probability and nondeterminism in a formal, unified way.

Findings

Unified formalism for Gaussian probability and nondeterminism

Application to noisy physical laws and open systems

Bridges categorical systems theory with probability theory

Abstract

We introduce categories of extended Gaussian maps and Gaussian relations which unify Gaussian probability distributions with relational nondeterminism in the form of linear relations. Both have crucial and well-understood applications in statistics, engineering, and control theory, but combining them in a single formalism is challenging. It enables us to rigorously describe a variety of phenomena like noisy physical laws, Willems' theory of open systems and uninformative priors in Bayesian statistics. The core idea is to formally admit vector subspaces $D \subseteq X$ as generalized uniform probability distribution. Our formalism represents a first bridge between the literature on categorical systems theory (signal-flow diagrams, linear relations, hypergraph categories) and notions of probability theory.