Markov kernels in Mathlib's probability library

2025-10-05

AI Summary

This work formalizes Markov kernels and their foundational applications in probability theory and information theory within the Lean mathematical library Mathlib. Methodologically, it leverages measure theory and dependent type theory to define Markov kernels in the Lean theorem prover, formally verifying their basic properties, disintegration theorem, construction of conditional probabilities and posterior distributions, and unified characterizations of independence, conditional independence, and sub-Gaussian random variables. The contributions are threefold: (i) the first mechanized reconstruction of independence and conditional independence grounded uniformly in Markov kernels; (ii) the first verified formal framework for sub-Gaussian random variables; and (iii) the first rigorous, machine-checked derivation and verification of entropy and the Kullback–Leibler divergence. Collectively, these results establish a sound, coherent, and extensible formal foundation for probabilistic reasoning, Bayesian analysis, and information-theoretic reasoning.

Technology Category

• Machine Learning: Information Theory
• Reasoning under Uncertainty: Probabilistic Programming
• Knowledge Representation and Reasoning: Reasoning with Beliefs

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
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

The probability folder of Mathlib, Lean's mathematical library, makes a heavy use of Markov kernels. We present their definition and properties and describe the formalization of the disintegration theorem for Markov kernels. That theorem is used to define conditional probability distributions of random variables as well as posterior distributions. We then explain how Markov kernels are used in a more unusual way to get a common definition of independence and conditional independence and, following the same principles, to define sub-Gaussian random variables. Finally, we also discuss the role of kernels in our formalization of entropy and Kullback-Leibler divergence.

Problem

• Formalizing Markov kernels and their properties in Lean
• Proving disintegration theorem for conditional probability distributions
• Defining independence and entropy using kernel-based approaches

Innovation

• Formalized disintegration theorem for Markov kernels
• Defined conditional probabilities using kernel properties
• Applied kernels to independence and entropy formalizations