Efficient Polynomial Representations and Transformations for Tractable Probabilistic Inference

Probabilistic circuits can compactly represent multilinear polynomials that encode probability distributions. This paper studies the relationships between various polynomial semantics, including likelihood, network, generating, and Fourier polynomials, and shows that they are all equivalent in the sense that any circuit for one can be transformed into a circuit for any of the others with only a polynomial increase in size. This establishes the tractability of marginal inference on the same class of distributions across these different polynomial representations.