  The sample complexity of simple binary hypothesis testing is the smallest
number of i.i.d. samples required to distinguish between two distributions $p$
and $q$ in either: (i) the prior-free setting, with type-I error at most
$\alpha$ and type-II error at most $\beta$; or (ii) the Bayesian setting, with
Bayes error at most $\delta$ and prior distribution $(\alpha, 1-\alpha)$. This
problem has only been studied when $\alpha = \beta$ (prior-free) or $\alpha =
1/2$ (Bayesian), and the sample complexity is known to be characterized by the
Hellinger divergence between $p$ and $q$, up to multiplicative constants. In
this paper, we derive a formula that characterizes the sample complexity (up to
multiplicative constants that are independent of $p$, $q$, and all error
parameters) for: (i) all $0 \le \alpha, \beta \le 1/8$ in the prior-free
setting; and (ii) all $\delta \le \alpha/4$ in the Bayesian setting. In
particular, the formula admits equivalent expressions in terms of certain
divergences from the Jensen--Shannon and Hellinger families. The main technical
result concerns an $f$-divergence inequality between members of the
Jensen--Shannon and Hellinger families, which is proved by a combination of
information-theoretic tools and case-by-case analyses. We explore applications
of our results to robust and distributed (locally-private and
communication-constrained) hypothesis testing.
