Testing Properties of Multiple Distributions with Few Samples

We propose a new setting for testing properties of distributions while receiving samples from several distributions, but few samples per distribution. Given samples from $s$ distributions, $p_1, p_2, ldots, p_s$, we design testers for the following problems: (1) Uniformity Testing: Testing whether all the $p_i$s are uniform or $epsilon$-far from being uniform in $ell_1$-distance (2) Identity Testing: Testing whether all the $p_i$s are equal to an explicitly given distribution $q$ or $epsilon$-far from $q$ in $ell_1$-distance, and (3) Closeness Testing: Testing whether all the $p_i$s are equal to a distribution $q$ which we have sample access to, or $epsilon$-far from $q$ in $ell_1$-distance. By assuming an additional natural condition about the source distributions, we provide sample optimal testers for all of these problems.