Polynomial Semantics of Tractable Probabilistic Circuits
CoRR(2024)
摘要
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.
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