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A Complete Diagrammatic Calculus for Conditional Gaussian Mixtures

Mateo Torres-Ruiz, Robin Piedeleu, Alexandra Silva, Fabio Zanasi

TL;DR

The paper develops a diagrammatic calculus for $CG$-mixtures that unifies discrete and Gaussian components within a symmetric monoidal category framework, enabling modular and compositional reasoning about conditional Gaussian mixtures. It defines a two-colour string-diagram syntax ($\mathsf{MixCirc}$) and a semantic target category ($\mathsf{MixGauss}$), proves a sound and complete equational theory $\mathsf{CGM}$ for diagrammatic equality, and provides normal-form techniques that guarantee canonic representations of CG-mixtures. The main results establish that any two circuits with the same semantics are inter-transformable using the axioms, supporting automatic reasoning about equivalence, marginalisation, and composition without explicit semantic computation. The framework lays groundwork for conditioning/inference and extends to broader mixture families, offering a formal, visual calculus for hybrid probabilistic models.

Abstract

We extend the synthetic theories of discrete and Gaussian categorical probability by introducing a diagrammatic calculus for reasoning about hybrid probabilistic models in which continuous random variables, conditioned on discrete ones, follow a multivariate Gaussian distribution. This setting includes important classes of models such as Gaussian mixture models, where each Gaussian component is selected according to a discrete variable. We develop a string diagrammatic syntax for expressing and combining these models, give it a compositional semantics, and equip it with a sound and complete equational theory that characterises when two models represent the same distribution.

A Complete Diagrammatic Calculus for Conditional Gaussian Mixtures

TL;DR

The paper develops a diagrammatic calculus for -mixtures that unifies discrete and Gaussian components within a symmetric monoidal category framework, enabling modular and compositional reasoning about conditional Gaussian mixtures. It defines a two-colour string-diagram syntax () and a semantic target category (), proves a sound and complete equational theory for diagrammatic equality, and provides normal-form techniques that guarantee canonic representations of CG-mixtures. The main results establish that any two circuits with the same semantics are inter-transformable using the axioms, supporting automatic reasoning about equivalence, marginalisation, and composition without explicit semantic computation. The framework lays groundwork for conditioning/inference and extends to broader mixture families, offering a formal, visual calculus for hybrid probabilistic models.

Abstract

We extend the synthetic theories of discrete and Gaussian categorical probability by introducing a diagrammatic calculus for reasoning about hybrid probabilistic models in which continuous random variables, conditioned on discrete ones, follow a multivariate Gaussian distribution. This setting includes important classes of models such as Gaussian mixture models, where each Gaussian component is selected according to a discrete variable. We develop a string diagrammatic syntax for expressing and combining these models, give it a compositional semantics, and equip it with a sound and complete equational theory that characterises when two models represent the same distribution.

Paper Structure

This paper contains 14 sections, 17 theorems, 16 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Background on (Categorical) Probability Theory
  3. Stochastic kernels
  4. Gaussian Probability
  5. Mixtures of Gaussians
  6. Conditional Gaussian mixtures
  7. String diagrammatic syntax
  8. Semantics
  9. Equational Theory
  10. Completeness
  11. Conclusions
  12. Appendix to Section \ref{['sec:background']}
  13. Appendix to Section \ref{['sec:axiomatisation']}
  14. Appendix to Section \ref{['sec:semantics']}

Key Result

Proposition 2

Mixtures of Gaussians $\sum_i p_i\cdot \mathcal{N}_{k}\left({\mu_i},{\Sigma_i}\right)$ are uniquely determined by their parameters, i.e., the mixture weights $p_i$, component means $\mu_i$, and covariances $\Sigma_i$.

Figures (2)

  • Figure 1: Semantics of circuit generators in terms of CG-mixtures.
  • Figure 2: Axioms of $\mathsf{CGM}$, the theory of Conditional Gaussian mixtures.

Theorems & Definitions (27)

  • Definition 1: Multivariate Gaussian distribution
  • Definition 2: Mixture distribution
  • Proposition 2
  • Definition 3: Conditional Gaussian mixture
  • Proposition 3
  • Proposition 3
  • Definition 4: $\mathsf{MixGauss}$
  • Remark 5
  • Proposition 6
  • Example 7
  • ...and 17 more