Property Testing of Joint Distributions using Conditional Samples

In this paper, we present the first non-trivial property tester for joint probability distributions in the recently introduced conditional sampling model. The conditional sampling framework provides an oracle for a distribution μ that takes as input a subset S of the domain Ω and returns a sample from the distribution μ conditioned on S.For a joint distribution of dimension n, we give a Õ(n3)-query uniformity tester, a Õ(n3)-query identity tester with a known distribution, and a Õ(n6)-query tester for testing independence of marginals. Our technique involves an elegant chain rule which can be proved using basic techniques of probability theory, yet powerful enough to avoid the curse of dimensionality. We also prove a sample complexity lower bound of Ω( 4 √ n) for testing uniformity of a joint distribution when the tester is only allowed to condition independently on the marginals. Our technique involves novel relations between Hellinger distance and total variational distance, and may be of independent interest.