  Given samples from an unknown multivariate distribution $p$, is it possible
to distinguish whether $p$ is the product of its marginals versus $p$ being far
from every product distribution? Similarly, is it possible to distinguish
whether $p$ equals a given distribution $q$ versus $p$ and $q$ being far from
each other? These problems of testing independence and goodness-of-fit have
received enormous attention in statistics, information theory, and theoretical
computer science, with sample-optimal algorithms known in several interesting
regimes of parameters. Unfortunately, it has also been understood that these
problems become intractable in large dimensions, necessitating exponential
sample complexity.
  Motivated by the exponential lower bounds for general distributions as well
as the ubiquity of Markov Random Fields (MRFs) in the modeling of
high-dimensional distributions, we initiate the study of distribution testing
on structured multivariate distributions, and in particular the prototypical
example of MRFs: the Ising Model. We demonstrate that, in this structured
setting, we can avoid the curse of dimensionality, obtaining sample and time
efficient testers for independence and goodness-of-fit. One of the key
technical challenges we face along the way is bounding the variance of
functions of the Ising model.
