Bayesian Networks and Proof-Nets: the proof-theory of Bayesian Inference

2026-02-03

AI Summary

This work establishes a formal proof-theoretic foundation for Bayesian inference and provides a graphical, compositional representation thereof. By forging a novel connection between Bayesian networks and proof nets from linear logic—inspired by the Curry–Howard correspondence—it develops a framework that unifies semantic rigor with computational efficiency. The approach introduces a flexible graph decomposition mechanism alongside a type inference system, enabling efficient and modular probabilistic reasoning. The primary contribution lies in formulating a proof-theoretic semantics for Bayesian inference, thereby offering a new theoretical toolkit for probabilistic programming and compositional reasoning.

Technology Category

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

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
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We study the correspondence between Bayesian Networks and graphical representation of proofs in linear logic. The goal of this paper is threefold: to develop a proof-theoretical account of Bayesian inference (in the spirit of the Curry-Howard correspondence between proofs and programs), to provide compositional graphical methods, and to take into account computational efficiency. We exploit the fact that the decomposition of a graph is more flexible than that of a proof-tree, or of a type-derivation, even if compositionality becomes more challenging.

Problem

• Bayesian Networks
• Proof-Nets
• Bayesian Inference
• Linear Logic
• Proof Theory

Innovation

• Bayesian Networks
• Proof-Nets
• Linear Logic
• Proof Theory
• Compositional Inference