Independence Testing for Bounded Degree Bayesian Networks

We study the following independence testing problem: given access to samples from a distribution P over ${0,1}^n$, decide whether P is a product distribution or whether it is ε-far in total variation distance from any product distribution. For arbitrary distributions, this problem requires exp(n) samples. We show in this work that if P has a sparse structure, then in fact only linearly many samples are required.Specifically, if P is Markov with respect to a Bayesian network whose underlying DAG has in-degree bounded by d, then ~Θ( $2^{d/2}$ ⋅n/ $ε^2$) samples are necessary and sufficient for independence testing.