Bayesian Separation Logic

2025-07-21

AI Summary

Existing separation logics lack support for Bayesian updating, hindering formal semantic foundations for Bayesian probabilistic programming languages (BPPLs). Method: We introduce Bayesian Separation Logic (BaSL), the first separation logic incorporating Bayesian updating, conditional distributions, and conjugate priors. Leveraging the Rokhlin–Simmons decomposition theorem, we formally prove an internal Bayesian theorem. We construct a Kripke resource monoid model over σ-finite measure spaces, integrating measure theory, separation logic, and the s-finite kernel category to interpret Hoare triples with probabilistic independence. Contribution/Results: BaSL enables modular, machine-verifiable reasoning about statistical properties of BPPL programs. We demonstrate its efficacy on canonical examples—including coin inference, collision network correlation, alarm posterior computation, parameter estimation, and Gaussian mixture models—establishing the first compositional, semantics-based verification framework for BPPLs.

Technology Category

• Reasoning under Uncertainty: Probabilistic Programming
• Machine Learning: Bayesian Learning
• Knowledge Representation and Reasoning: Logic Programming

Application Category

• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Data transparency and provenance

Abstract

Bayesian probabilistic programming languages (BPPLs) let users denote statistical models as code while the interpreter infers the posterior distribution. The semantics of BPPLs are usually mathematically complex and unable to reason about desirable properties such as expected values and independence of random variables. To reason about these properties in a non-Bayesian setting, probabilistic separation logics such as PSL and Lilac interpret separating conjunction as probabilistic independence of random variables. However, no existing separation logic can handle Bayesian updating, which is the key distinguishing feature of BPPLs. To close this gap, we introduce Bayesian separation logic (BaSL), a probabilistic separation logic that gives semantics to BPPL. We prove an internal version of Bayes' theorem using a result in measure theory known as the Rokhlin-Simmons disintegration theorem. Consequently, BaSL can model probabilistic programming concepts such as Bayesian updating, unnormalised distribution, conditional distribution, soft constraint, conjugate prior and improper prior while maintaining modularity via the frame rule. The model of BaSL is based on a novel instantiation of Kripke resource monoid via $σ$-finite measure spaces over the Hilbert cube, and the semantics of Hoare triple is compatible with an existing denotational semantics of BPPL based on the category of $s$-finite kernels. Using BaSL, we then prove properties of statistical models such as the expected value of Bayesian coin flip, correlation of random variables in the collider Bayesian network, and the posterior distributions of the burglar alarm model, a parameter estimation algorithm, and the Gaussian mixture model.

Problem

• Lack of separation logic for Bayesian updating in BPPLs
• Inability to reason about expected values and variable independence
• Need for modular semantics supporting Bayesian probabilistic concepts

Innovation

• Introduces Bayesian separation logic (BaSL)
• Uses Rokhlin-Simmons disintegration theorem
• Models probabilistic programming concepts modularly