  Generating functions, which are widely used in combinatorics and probability
theory, encode function values into the coefficients of a polynomial. In this
paper, we explore their use as a tractable probabilistic model, and propose
probabilistic generating circuits (PGCs) for their efficient representation.
PGCs strictly subsume many existing tractable probabilistic models, including
determinantal point processes (DPPs), probabilistic circuits (PCs) such as
sum-product networks, and tractable graphical models. We contend that PGCs are
not just a theoretical framework that unifies vastly different existing models,
but also show huge potential in modeling realistic data. We exhibit a simple
class of PGCs that are not trivially subsumed by simple combinations of PCs and
DPPs, and obtain competitive performance on a suite of density estimation
benchmarks. We also highlight PGCs' connection to the theory of strongly
Rayleigh distributions.
