  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 (flipping each bit independently with some constant probability $q\in
[0,1/2)$). In the absence of noise, predictive learning on Ising models was
recently studied by Bresler and Karzand (2020); this paper quantifies how noise
in the hidden model impacts the tasks of structure recovery and marginal
distribution estimation by proving upper and lower bounds on the sample
complexity. Our results generalize state-of-the-art bounds reported in prior
work, and they exactly recover the noiseless case ($q=0$). In fact, for any
tree with $p$ vertices and probability of incorrect recovery $\delta>0$, the
sufficient number of samples remains logarithmic as in the noiseless case,
i.e., $\mathcal{O}(\log(p/\delta))$, while the dependence on $q$ is
$\mathcal{O}\big( 1/(1-2q)^{4} \big)$, 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.
