  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.
