Problem 66: Distinguishing Distributions with Conditional Samples

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Suggested by Eldar Fischer
Source Bertinoro 2014
Short link https://sublinear.info/66

Suppose we are given access to two distributions $P$ and $Q$ over $\{1,2, \ldots, n\}$ and wish to test if they are the same or are at least $\epsilon$ apart under the $\ell_1$ distance. Assume that we have access to conditional samples: a query consists of a set $S \subseteq \{1,2, \ldots, n\}$ and the output is a sample drawn from the conditional distribution on $A$. In other words, a conditional sample on $P$ given $S$ is drawn from the distribution where

$$ \text{Pr}(j) = \begin{cases} \frac{p_j}{\sum_{i \in A} p_i} & j \in A \\ 0 & \text{otherwise} \end{cases} $$

It is known that if one of the distributions is fixed, then a constant number ($f(1/\epsilon)$, where $f$ is an arbitrary function) of queries suffice to test the distributions.

What can we say if both distributions are unknown?