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 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 by Bayes nets of bounded 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.

翻译：在本文中，我们建立了全变差距离估计与概率推断之间一种新颖的联系。具体而言，我们提出了一种高效且保持结构约化的方法，将全变差距离的相对近似问题转化为有向图模型上的概率推断问题。这一约化导致了一个完全多项式随机近似方案，用于估计任意一类贝叶斯网络分布之间的全变差距离，前提是该类贝叶斯网络存在高效的概率推断算法。特别地，它针对由有界树宽的贝叶斯网络定义的分布之间的全变差距离，给出了一个完全多项式随机近似方案。在此之前，这类近似方案仅存在于估计乘积分布之间的全变差距离。我们的方法引入了一种高维分布的$partial$耦合概念，这可能具有独立的研究意义。