### Scaling Integer Arithmetic in Probabilistic Programs

*Proceedings of the 39th Conference on Uncertainty in Artificial Intelligence* (UAI 2023), PMLR 216:260-270.

William X. Cao, Poorva Garg, Ryan Tjoa, Steven Holtzen, Todd Millstein, Guy Van den Broeck

Distributions on integers are ubiquitous in probabilistic modeling but remain challenging for many of today's probabilistic programming languages (PPLs). The core challenge comes from discrete structure: many of today's PPL inference strategies rely on enumeration, sampling, or differentiation in order to scale, which fail for high-dimensional complex discrete distributions involving integers. Our insight is that there is structure in arithmetic that these approaches are not using. We present a binary encoding strategy for discrete distributions that exploits the rich logical structure of integer operations like summation and comparison. We leverage this structured encoding with knowledge compilation to perform exact probabilistic inference, and show that this approach scales to much larger integer distributions with arithmetic.

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