The paper studies the relationships between various polynomial representations of probability distributions that can be compactly encoded using probabilistic circuits (PCs). These polynomial representations include:
The key findings are:
For binary distributions, the authors prove that each of these polynomial representations is equivalent in the sense that a circuit for one can be transformed into a circuit for any of the others with only a polynomial increase in size. This implies that they all support tractable marginal inference on the same class of distributions.
The authors provide linear-time inference algorithms for circuits computing likelihood polynomials, and polynomial-time transformations between the different polynomial representations.
The authors explore the natural extension of generating polynomials to categorical distributions and show that for categories with 4 or more values, inference becomes #P-hard.
The authors also discuss how the transformations simplify when the circuits are constrained to be decomposable, a common structural property that guarantees tractable marginal inference.
Overall, the paper establishes the equivalence of these polynomial representations, unifying various tractable probabilistic models and enabling the transfer of theoretical and algorithmic results between them.
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by Oliver Broad... at arxiv.org 04-30-2024
https://arxiv.org/pdf/2402.09085.pdfDeeper Inquiries