Probabilistic circuits compute multilinear polynomials that represent probability distributions. They are tractable models that support efficient marginal inference. However, various polynomial semantics have been considered in the literature (e.g., network polynomials, likelihood polynomials, generating functions, Fourier transforms, and characteristic polynomials). The relationships between these polynomial encodings of distributions is largely unknown. In this paper, we prove that for binary distributions, each of these probabilistic circuit models is equivalent in the sense that any circuit for one of them can be transformed into a circuit for any of the others with only a polynomial increase in size. They are therefore all tractable for marginal inference on the same class of distributions. Finally, we explore the natural extension of one such polynomial semantics, called probabilistic generating circuits, to categorical random variables, and establish that marginal inference becomes #P-hard.

翻译：概率电路计算表示概率分布的多线性多项式。它们是支持高效边缘推断的可处理模型。然而，文献中已考虑了多种多项式语义（例如网络多项式、似然多项式、生成函数、傅里叶变换和特征多项式）。这些分布的多项式编码之间的关系在很大程度上是未知的。本文证明，对于二元分布，每种概率电路模型在如下意义上是等价的：针对其中任一模型的电路，都能仅以多项式规模增长转化为其他任一模型的电路。因此，这些模型在相同分布类上的边缘推断中均为可处理模型。最后，我们探索了其中一种多项式语义（称为概率生成电路）向分类随机变量的自然扩展，并证实此时边缘推断变为#P-难问题。