On Approximating Total Variation Distance

Total variation distance (TV distance) is a fundamental notion of distance between probability distributions. In this work, we introduce and study the computational problem of determining the TV distance between two product distributions over the domain $\{0,1\}^n$. We establish the following results. 1. Exact computation of TV distance between two product distributions is $\#\mathsf{P}$-complete. This is in stark contrast with other distance measures such as KL, Chi-square, and Hellinger which tensorize over the marginals. 2. Given two product distributions $P$ and $Q$ with marginals of $P$ being at least $1/2$ and marginals of $Q$ being at most the respective marginals of $P$, there exists a fully polynomial-time randomized approximation scheme (FPRAS) for computing the TV distance between $P$ and $Q$. In particular, this leads to an efficient approximation scheme for the interesting case when $P$ is an arbitrary product distribution and $Q$ is the uniform distribution. We pose the question of characterizing the complexity of approximating the TV distance between two arbitrary product distributions as a basic open problem in computational statistics.