  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 are strictly more expressive efficient than 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 great 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.
