Exact Bayesian Inference on Discrete Models via Probability Generating Functions: A Probabilistic Programming Approach

We present an exact Bayesian inference method for discrete statistical models, which can find exact solutions to many discrete inference problems, even with infinite support and continuous priors. To express such models, we introduce a probabilistic programming language that supports discrete and continuous sampling, discrete observations, affine functions, (stochastic) branching, and conditioning on events. Our key tool is probability generating functions: they provide a compact closed-form representation of distributions that are definable by programs, thus enabling the exact computation of posterior probabilities, expectation, variance, and higher moments. Our inference method is provably correct, fully automated and uses automatic differentiation (specifically, Taylor polynomials), but does not require computer algebra. Our experiments show that its performance on a range of real-world examples is competitive with approximate Monte Carlo methods, while avoiding approximation errors.