  Given a mixture between two populations of coins, "positive" coins that each
have -- unknown and potentially different -- bias $\geq\frac{1}{2}+\Delta$ and
"negative" coins with bias $\leq\frac{1}{2}-\Delta$, we consider the task of
estimating the fraction $\rho$ of positive coins to within additive error
$\epsilon$. We achieve an upper and lower bound of
$\Theta(\frac{\rho}{\epsilon^2\Delta^2}\log\frac{1}{\delta})$ samples for a
$1-\delta$ probability of success, where crucially, our lower bound applies to
all fully-adaptive algorithms. Thus, our sample complexity bounds have tight
dependence for every relevant problem parameter. A crucial component of our
lower bound proof is a decomposition lemma (see Lemmas 17 and 18) showing how
to assemble partially-adaptive bounds into a fully-adaptive bound, which may be
of independent interest: though we invoke it for the special case of Bernoulli
random variables (coins), it applies to general distributions. We present
simulation results to demonstrate the practical efficacy of our approach for
realistic problem parameters for crowdsourcing applications, focusing on the
"rare events" regime where $\rho$ is small. The fine-grained adaptive flavor of
both our algorithm and lower bound contrasts with much previous work in
distributional testing and learning.
