Predictive Learning on Sign-Valued Hidden Markov Trees

We provide high-probability sample complexity guarantees for exact structure recovery and accurate Predictive Learning using noise-corrupted samples from an acyclic (tree-shaped) graphical model. The hidden variables follow a tree-structured Ising model distribution whereas the observable variables are generated by a binary symmetric channel, taking the hidden variables as its input. This model arises naturally in a variety of applications, such as in physics, biology, computer science, and finance. The noiseless structure learning problem has been studied earlier by Bresler and Karzand (2018); this paper quantifies how noise in the hidden model impacts the sample complexity of structure learning and predictive distributional inference by proving upper and lower bounds on the sample complexity. Quite remarkably, for any tree with $p$ vertices and probability of incorrect recovery $\delta>0$, the order of necessary number of samples remains logarithmic as in the noiseless case, i.e., $\mathcal{O}(\log(p/\delta))$, for both aforementioned tasks. We also present a new equivalent of Isserlis' Theorem for sign-valued tree-structured distributions, yielding a new low-complexity algorithm for higher order moment estimation.