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