This work initiates a systematic investigation of testing {\em high-dimensional} structured distributions by focusing on testing {\em 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 {\em sublinear} in the dimension and are sample-optimal, up to constant factors.
['Clément L. Canonne', 'Ilias Diakonikolas', 'Daniel M. Kane', 'Alistair Stewart']
Testing Bayesian Networks