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 $\mathcal{S}$ of a given size, of the total variation between the marginals of P and Q on $\mathcal{S}$. 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 $\beta$, we prove that $\max\{e^{2\beta}\log p, \eta^{-2}\log(p/\eta)\} $ i.i.d. samples suffices to get a distribution (from the same class) with ssTV at most $\eta$ from the one generating the samples.