# Measure Theoretic Weighted Model Integration

arXiv: https://arxiv.org/abs/2103.13901 · HTML: https://mlsanity.com/paper/2103.13901/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 6 of its 29 references that are in the map.
- First posted on arXiv 25 Mar 2021.

## Builds on (most cited first, 6 of 6)

- [A Convenient Category for Higher-Order Probability Theory](https://mlsanity.com/paper/1701.02547/) (2017)
- [Efficient Search-Based Weighted Model Integration](https://mlsanity.com/paper/1903.05334/) (2019)
- [Scaling up Hybrid Probabilistic Inference with Logical and Arithmetic Constraints via Message Passing](https://mlsanity.com/paper/2003.00126/) (2020)
- [Probabilistic Inference Modulo Theories](https://mlsanity.com/paper/1605.08367/) (2016)
- [Discrete-Continuous Mixtures in Probabilistic Programming: Generalized Semantics and Inference Algorithms](https://mlsanity.com/paper/1806.02027/) (2018)
- [On Lebesgue Integral Quadrature](https://mlsanity.com/paper/1807.06007/) (2018)

## Built on by (most cited first, 1 of 1)

- [Semirings for Probabilistic and Neuro-Symbolic Logic Programming](https://mlsanity.com/paper/2402.13782/) (2024)

## Abstract (arXiv)

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.