  We study the fundamental problems of identity testing (goodness of fit), and
closeness testing (two sample test) of distributions over $k$ elements, under
differential privacy. While the problems have a long history in statistics,
finite sample bounds for these problems have only been established recently.
  In this work, we derive upper and lower bounds on the sample complexity of
both the problems under $(\varepsilon, \delta)$-differential privacy. We
provide optimal sample complexity algorithms for identity testing problem for
all parameter ranges, and the first results for closeness testing. Our
closeness testing bounds are optimal in the sparse regime where the number of
samples is at most $k$.
  Our upper bounds are obtained by privatizing non-private estimators for these
problems. The non-private estimators are chosen to have small sensitivity. We
propose a general framework to establish lower bounds on the sample complexity
of statistical tasks under differential privacy. We show a bound on
differentially private algorithms in terms of a coupling between the two
hypothesis classes we aim to test. By constructing carefully chosen priors over
the hypothesis classes, and using Le Cam's two point theorem we provide a
general mechanism for proving lower bounds. We believe that the framework can
be used to obtain strong lower bounds for other statistical tasks under
privacy.
