  This work initiates a systematic investigation of testing high-dimensional
structured distributions by focusing on testing Bayesian networks -- the
prototypical family of directed graphical models. A Bayesian network is defined
by a directed acyclic graph, where we associate a random variable with each
node. The value at any particular node is conditionally independent of all the
other non-descendant nodes once its parents are fixed. Specifically, we study
the properties of identity testing and closeness testing of Bayesian networks.
Our main contribution is the first non-trivial efficient testing algorithms for
these problems and corresponding information-theoretic lower bounds. For a wide
range of parameter settings, our testing algorithms have sample complexity
sublinear in the dimension and are sample-optimal, up to constant factors.
