Hamiltonian Monte Carlo for Probabilistic Programs with Discontinuities
This addresses a bottleneck for users of probabilistic programming languages who need to handle non-differentiable models, though it is incremental as it builds on existing HMC techniques.
The paper tackles the problem of performing statistical inference in probabilistic programs with discontinuities, which is challenging for Hamiltonian Monte Carlo (HMC) due to its reliance on derivatives, and presents a method using extensions to HMC and a restricted language (SPPL) to enable inference while preserving the interpretation of if-else statements.
Hamiltonian Monte Carlo (HMC) is arguably the dominant statistical inference algorithm used in most popular "first-order differentiable" Probabilistic Programming Languages (PPLs). However, the fact that HMC uses derivative information causes complications when the target distribution is non-differentiable with respect to one or more of the latent variables. In this paper, we show how to use extensions to HMC to perform inference in probabilistic programs that contain discontinuities. To do this, we design a Simple first-order Probabilistic Programming Language (SPPL) that contains a sufficient set of language restrictions together with a compilation scheme. This enables us to preserve both the statistical and syntactic interpretation of if-else statements in the probabilistic program, within the scope of first-order PPLs. We also provide a corresponding mathematical formalism that ensures any joint density denoted in such a language has a suitably low measure of discontinuities.