The idea of probability generating function achieving similar forms of "conjugacy" as Gaussian random variables and (finite) discrete random variables is very interesting. This enables a fast and exact inference algorithm on Poisson latent variable models. The formulation involving probability generating function does not seem to be constrained to Poisson random variables but all the simulations and real application pertain to Poisson HMM. It is unclear if there's anything special with Poisson that enables better parameter estimates. It is also unclear if there are other Poisson latent variable models other than Poisson HMMs with wide applications.

Graphical models with latent count variables arise in a number of fields. Standard exact inference techniques such as variable elimination and belief propagation do not apply to these models because the latent variables have countably infinite support. As a result, approximations such as truncation or MCMC are employed. We present the first exact inference algorithms for a class of models with latent count variables by developing a novel representation of countably infinite factors as probability generating functions, and then performing variable elimination with generating functions. Our approach is exact, runs in pseudo-polynomial time, and is much faster than existing approximate techniques.