Markov kernels in Mathlib's probability library
This work provides a foundational formalization for probability theory in the Lean theorem prover, benefiting researchers using formal verification for probabilistic reasoning.
The paper presents the formalization of Markov kernels in Mathlib, including the disintegration theorem for conditional probability and posterior distributions, and their use in defining independence, sub-Gaussian random variables, and information-theoretic quantities. No concrete performance numbers are given.
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.