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.

翻译：整数上的分布在概率建模中无处不在，但对于当今的许多概率编程语言（PPL）而言，它们仍然具有挑战性。核心挑战源于离散结构：当今许多 PPL 推理策略依赖枚举、采样或微分来实现规模化，但对于涉及整数的高维复杂离散分布，这些方法往往失效。我们的洞察在于，算术中存在这些方法未利用的结构。我们提出了一种针对离散分布的二进制编码策略，该策略利用了整数运算（如求和与比较）丰富的逻辑结构。我们借助知识编译来利用这种结构化编码执行精确概率推理，并证明该方法能够显著扩展到包含算术运算的更大整数分布。