  We design efficient distance approximation algorithms for several classes of
structured high-dimensional distributions. Specifically, we show algorithms for
the following problems:
  - Given sample access to two Bayesian networks $P_1$ and $P_2$ over known
directed acyclic graphs $G_1$ and $G_2$ having $n$ nodes and bounded in-degree,
approximate $d_{tv}(P_1,P_2)$ to within additive error $\epsilon$ using
$poly(n,\epsilon)$ samples and time
  - Given sample access to two ferromagnetic Ising models $P_1$ and $P_2$ on
$n$ variables with bounded width, approximate $d_{tv}(P_1, P_2)$ to within
additive error $\epsilon$ using $poly(n,\epsilon)$ samples and time
  - Given sample access to two $n$-dimensional Gaussians $P_1$ and $P_2$,
approximate $d_{tv}(P_1, P_2)$ to within additive error $\epsilon$ using
$poly(n,\epsilon)$ samples and time
  - Given access to observations from two causal models $P$ and $Q$ on $n$
variables that are defined over known causal graphs, approximate $d_{tv}(P_a,
Q_a)$ to within additive error $\epsilon$ using $poly(n,\epsilon)$ samples,
where $P_a$ and $Q_a$ are the interventional distributions obtained by the
intervention $do(A=a)$ on $P$ and $Q$ respectively for a particular variable
$A$.
  Our results are the first efficient distance approximation algorithms for
these well-studied problems. They are derived using a simple and general
connection to distribution learning algorithms. The distance approximation
algorithms imply new efficient algorithms for {\em tolerant} testing of
closeness of the above-mentioned structured high-dimensional distributions.
