Measure Theoretic Weighted Model Integration
This work provides a more principled theoretical framework for probabilistic inference in hybrid domains, addressing a foundational issue for researchers in AI and machine learning, though it is incremental in nature.
The paper tackles the theoretical foundation of weighted model integration (WMI) by proposing a measure-theoretic formulation using Lebesgue integration, which unifies discrete and continuous variables and naturally reduces to weighted model counting (WMC) as a special case.
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.