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 ⊆ [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 μ; testing whether D = D∗ for an explicitly provided D∗; testing whether two unknown distributions D₁ and D₂ 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 Ω(√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/ε)-query algorithms (and in some cases poly(1/ε)-query algorithms independent of N) for all these problems in our conditional sampling setting.