Testing probability distributions using conditional samples

Abstract

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 $S \subseteq [N]$ 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 $\mathcal{U}$; testing whether $D = D^\ast$ for an explicitly provided $D^\ast$; 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 $\Omega(\sqrt{N})$ sample complexity in the standard model (and in some cases the complexity must be almost linear in $N$), we give $\mathrm{poly}(\log N, 1/\varepsilon)$-query algorithms (and in some cases $\mathrm{poly}(1/\varepsilon)$-query algorithms independent of $N$) for all these problems in our conditional sampling setting. *Independently from our work, Chakraborty et al. also considered this framework. We discuss their work in Subsection [1.4].Comment: Significant changes on Section 9 (detailing and expanding the proof of Theorem 16). Several clarifications and typos fixed in various place