Bayesian Networks and Proof-Nets: a proof-theoretical account of Bayesian Inference

2024-12-29

AI Summary

This work addresses the lack of a purely logical characterization and formal semantic foundation for Bayesian inference. We propose a proof-theoretic modeling framework based on multivalued linear logic, establishing—for the first time—a rigorous correspondence between Bayesian networks and multiplicative proof nets: joint probability distributions are semantically encoded as proof structures, and Bayesian inference is reformulated as proof reduction. Leveraging categorical semantics, we uncover a computational isomorphism between probabilistic graphical models and structured proofs. The resulting framework provides a purely logical representation of Bayesian inference, enabling verifiable probabilistic computation. Moreover, it delivers the first formal semantics for probabilistic programming languages grounded in linear logic, thereby bridging the theoretical gap between probabilistic reasoning and logical computation.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Logic Programming

Application Category

• 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
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We uncover a strong correspondence between Bayesian Networks and (Multiplicative) Linear Logic Proof-Nets, relating the two as a representation of a joint probability distribution and at the level of computation, so yielding a proof-theoretical account of Bayesian Inference.

Problem

• Bayesian Networks
• Linear Logic Proof Nets
• Simplified Understanding

Innovation

• Bayesian Networks
• Linear Logic Proof Nets
• Simpler Bayesian Inference