A Bunched Logic for Conditional Independence

TL;DR

This paper introduces a new logical framework based on bunched implications to reason about conditional independence in probabilistic programs, providing models that capture both probabilistic and non-probabilistic notions and a logic for verification.

Contribution

It extends bunched implications logic with non-commutative conjunctions, models conditional independence using Markov kernels, and develops a program logic for verification.

Findings

Logical formula for conditional independence in Markov kernel model

Model capturing join dependency via powerset monad

Program logic for verifying conditional independence

Abstract

Independence and conditional independence are fundamental concepts for reasoning about groups of random variables in probabilistic programs. Verification methods for independence are still nascent, and existing methods cannot handle conditional independence. We extend the logic of bunched implications (BI) with a non-commutative conjunction and provide a model based on Markov kernels; conditional independence can be directly captured as a logical formula in this model. Noting that Markov kernels are Kleisli arrows for the distribution monad, we then introduce a second model based on the powerset monad and show how it can capture join dependency, a non-probabilistic analogue of conditional independence from database theory. Finally, we develop a program logic for verifying conditional independence in probabilistic programs.