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A Complete Diagrammatic Calculus for Conditional Gaussian Mixtures

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

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Subject Classification

ACM Subject Classification
  • Theory of computation → Categorical semantics
  • Theory of computation → Logic
Keywords
  • String diagrams
  • Category theory
  • Mixture models
  • Probability theory

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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 families of distributions such as Gaussian mixtures, where each Gaussian component is selected according to a discrete variable. We develop a string diagrammatic syntax for distributions of this type, give it a compositional semantics, and equip it with a sound and complete equational theory that characterises when two mixtures represent the same distribution.

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Mateo Torres-Ruiz, Robin Piedeleu, Alexandra Silva, and Fabio Zanasi. A Complete Diagrammatic Calculus for Conditional Gaussian Mixtures. In 34th EACSL Annual Conference on Computer Science Logic (CSL 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 363, pp. 11:1-11:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.CSL.2026.11

Author Details

Mateo Torres-Ruiz
  • University College London, UK
Robin Piedeleu
  • University College London, UK
Alexandra Silva
  • Cornell University, Ithaca, NY, USA
Fabio Zanasi
  • University College London, UK

Funding

This work was partially supported by ARIA’s Safeguarded AI programme. RP, MTR and AS acknowledge support from erc grant Autoprobe no. 101002697.

Acknowledgements

The authors would like to thank the anonymous reviewers for their helpful suggestions.

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