Hybrid Probabilistic Inference with Logical Constraints: Tractability and Message Passing

TL;DR

This paper introduces a message passing approach for hybrid probabilistic inference using weighted model integration, achieving linear-time computation of marginals and establishing structural conditions for tractability.

Contribution

It presents a novel MI formulation with message passing, enabling efficient inference and characterizing the structural bounds for tractability in hybrid probabilistic models.

Findings

Linear-time marginal density computation via message passing.

Structural bounds on primal graph for tractable inference.

Theoretical proof of necessity and sufficiency of bounds.

Abstract

Weighted model integration (WMI) is a very appealing framework for probabilistic inference: it allows to express the complex dependencies of real-world hybrid scenarios where variables are heterogeneous in nature (both continuous and discrete) via the language of Satisfiability Modulo Theories (SMT); as well as computing probabilistic queries with arbitrarily complex logical constraints. Recent work has shown WMI inference to be reducible to a model integration (MI) problem, under some assumptions, thus effectively allowing hybrid probabilistic reasoning by volume computations. In this paper, we introduce a novel formulation of MI via a message passing scheme that allows to efficiently compute the marginal densities and statistical moments of all the variables in linear time. As such, we are able to amortize inference for arbitrarily rich MI queries when they conform to the problem structure, here represented as the primal graph associated to the SMT formula. Furthermore, we theoretically trace the tractability boundaries of exact MI. Indeed, we prove that in terms of the structural requirements on the primal graph that make our MI algorithm tractable - bounding its diameter and treewidth - the bounds are not only sufficient, but necessary for tractable inference via MI.