Exact Probabilistic Inference Using Generating Functions

TL;DR

This paper explores the use of generating functions for exact probabilistic inference in probabilistic programming, addressing challenges like recursion, infinite support, and conditioning to improve precision and efficiency.

Contribution

It introduces a novel approach leveraging generating functions to perform exact inference in probabilistic programs, especially handling conditioning and recursion.

Findings

Generating functions enable exact inference in probabilistic programs.

The approach effectively handles conditioning and recursion.

Potential for improved precision and efficiency in probabilistic reasoning.

Abstract

Probabilistic programs are typically normal-looking programs describing posterior probability distributions. They intrinsically code up randomized algorithms and have long been at the heart of modern machine learning and approximate computing. We explore the theory of generating functions [19] and investigate its usage in the exact quantitative reasoning of probabilistic programs. Important topics include the exact representation of program semantics [13], proving exact program equivalence [5], and -- as our main focus in this extended abstract -- exact probabilistic inference. In probabilistic programming, inference aims to derive a program's posterior distribution. In contrast to approximate inference, inferring exact distributions comes with several benefits [8], e.g., no loss of precision, natural support for symbolic parameters, and efficiency on models with certain structures. Exact probabilistic inference, however, is a notoriously hard task [6,12,17,18]. The challenges mainly arise from three program constructs: (1) unbounded while-loops and/or recursion, (2) infinite-support distributions, and (3) conditioning (via posterior observations). We present our ongoing research in addressing these challenges (with a focus on conditioning) leveraging generating functions and show their potential in facilitating exact probabilistic inference for discrete probabilistic programs.