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 U ; 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(logN, 1/ )-query algorithms (and in some cases poly(1/ )-query algorithms independent of N) for all these problems in our conditional sampling setting. ∗[email protected], Columbia University. Supported by NSF grants CCF-1115703 and CCF-1319788. †[email protected], Tel Aviv University. Supported by ISF grants 246/08 and 671/13. ‡[email protected], Columbia University. Supported by NSF grants CCF-0915929 and CCF-1115703.