Measure Theoretic Weighted Model Integration

TL;DR

This paper introduces a measure theoretic framework for weighted model integration (WMI), unifying discrete and continuous variables under a rigorous mathematical foundation, and generalizing weighted model counting (WMC).

Contribution

It provides a principled measure theoretic formulation of WMI that naturally reduces to WMC when continuous variables are absent, positioning WMC as a special case of WMI.

Findings

The measure theoretic approach unifies discrete and continuous variables.

WMC is shown to be a special case of the proposed WMI framework.

The formulation is mathematically rigorous and grounded in Lebesgue integration.

Abstract

Weighted model counting (WMC) is a popular framework to perform probabilistic inference with discrete random variables. Recently, WMC has been extended to weighted model integration (WMI) in order to additionally handle continuous variables. At their core, WMI problems consist of computing integrals and sums over weighted logical formulas. From a theoretical standpoint, WMI has been formulated by patching the sum over weighted formulas, which is already present in WMC, with Riemann integration. A more principled approach to integration, which is rooted in measure theory, is Lebesgue integration. Lebesgue integration allows one to treat discrete and continuous variables on equal footing in a principled fashion. We propose a theoretically sound measure theoretic formulation of weighted model integration, which naturally reduces to weighted model counting in the absence of continuous variables. Instead of regarding weighted model integration as an extension of weighted model counting, WMC emerges as a special case of WMI in our formulation.