  Probabilistic circuits compute multilinear polynomials that represent
multivariate 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, and Fourier transforms). The relationships
between circuit representations of these polynomial encodings of distributions
is largely unknown. In this paper, we prove that for distributions over binary
variables, 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 inference becomes #P-hard.
