Testing Probability Distributions using Conditional Samples.

We study a new framework for property testing of probability distributions, by considering distribution testing algorithms that have access to a conditional sampling oracle. This is an oracle that takes as input a subset S subset of [N] of the domain [N] of the unknown probability distribution D and returns a draw from the conditional probability distribution D restricted to S. This new model allows considerable flexibility in the design of distribution testing algorithms; in particular, testing algorithms in this model can be adaptive. We study a wide range of natural distribution testing problems in this new framework and some of its variants, giving both upper and lower bounds on query complexity. These problems include testing whether D is the uniform distribution U; testing whether D = D* for an explicitly provided D*; testing whether two unknown distributions D-1 and D-2 are equivalent; and estimating the variation distance between D and the uniform distribution. At a high level, our main finding is that the new conditional sampling framework we consider is a powerful one: while all the problems mentioned above have Omega(root N) sample complexity in the standard model (and in some cases the complexity must be almost linear in N), we give poly(log N, 1/epsilon)-query algorithms (and in some cases poly(1/epsilon)-query algorithms independent of N) for all these problems in our conditional sampling setting.