Total Variation Distance Meets Probabilistic Inference

TL;DR

This paper introduces a new method connecting total variation distance estimation with probabilistic inference, enabling efficient approximation schemes for complex distributions over Bayesian networks with small treewidth.

Contribution

It provides the first FPRAS for TV distance estimation between distributions over Bayes nets with small treewidth, extending beyond product distributions.

Findings

Develops a reduction from TV distance approximation to probabilistic inference.

Introduces a new concept of partial couplings for high-dimensional distributions.

Achieves an efficient approximation scheme for distributions over small-treewidth Bayes nets.

Abstract

In this paper, we establish a novel connection between total variation (TV) distance estimation and probabilistic inference. In particular, we present an efficient, structure-preserving reduction from relative approximation of TV distance to probabilistic inference over directed graphical models. This reduction leads to a fully polynomial randomized approximation scheme (FPRAS) for estimating TV distances between same-structure distributions over any class of Bayes nets for which there is an efficient probabilistic inference algorithm. In particular, it leads to an FPRAS for estimating TV distances between distributions that are defined over a common Bayes net of small treewidth. Prior to this work, such approximation schemes only existed for estimating TV distances between product distributions. Our approach employs a new notion of $partial$ couplings of high-dimensional distributions, which might be of independent interest.