Learning a Tree-Structured Ising Model in Order to Make Predictions

We study the problem of learning a tree graphical model from samples such that low-order marginals are accurate. We define a distance (“small set TV” or ssTV) between distributions P and Q by taking the maximum, over all subsets S of a given size, of the total variation between the marginals of P and Q on S. Approximating a distribution to within small ssTV allows making predictions based on partial observations. Focusing on pairwise marginals and tree-structured Ising models on p nodes with maximum edge strength β, we prove that max{e2β log p, η−2 log(p/η)} i.i.d. samples suffices to get a distribution (from the same class) with ssTV at most η from the one generating the samples.