A Complete Diagrammatic Calculus for Conditional Gaussian Mixtures

2025-10-06

AI Summary

Modeling and equivalence reasoning for discrete-continuous hybrid probabilistic models—such as conditional Gaussian mixture models (CGMMs), where continuous variables follow multivariate Gaussians conditioned on discrete variables—remains challenging due to the lack of compositional, syntactic, and semantic foundations. Method: We introduce the first complete string diagram calculus for CGMMs, integrating categorical probability theory and compositional semantics to yield a graphical syntax with rigorous denotational meaning, accompanied by a sound and complete equational theory. Contribution/Results: This is the first framework to algebraically compose, visually represent, and precisely decide structural equivalence for such models: two diagrammatic expressions are equivalent if and only if they induce identical probability distributions. The calculus supports model construction, decomposition, optimization, and formal verification, thereby establishing a novel formal foundation for probabilistic programming and causal modeling.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Reasoning under Uncertainty: Probabilistic Programming
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Developing diagrammatic calculus for hybrid probabilistic models
• Modeling continuous variables conditioned on discrete variables
• Providing sound equational theory for distribution equivalence

Innovation

• Diagrammatic calculus for hybrid probabilistic models
• String diagram syntax with compositional semantics
• Complete equational theory for distribution equivalence