  Given $n$ observations from two balanced classes, consider the task of
labeling an additional $m$ inputs that are known to all belong to \emph{one} of
the two classes. Special cases of this problem are well-known: with complete
knowledge of class distributions ($n=\infty$) the problem is solved optimally
by the likelihood-ratio test; when $m=1$ it corresponds to binary
classification; and when $m\approx n$ it is equivalent to two-sample testing.
The intermediate settings occur in the field of likelihood-free inference,
where labeled samples are obtained by running forward simulations and the
unlabeled sample is collected experimentally. In recent work it was discovered
that there is a fundamental trade-off between $m$ and $n$: increasing the data
sample $m$ reduces the amount $n$ of training/simulation data needed. In this
work we (a) introduce a generalization where unlabeled samples come from a
mixture of the two classes -- a case often encountered in practice; (b) study
the minimax sample complexity for non-parametric classes of densities under
\textit{maximum mean discrepancy} (MMD) separation; and (c) investigate the
empirical performance of kernels parameterized by neural networks on two tasks:
detection of the Higgs boson and detection of planted DDPM generated images
amidst CIFAR-10 images. For both problems we confirm the existence of the
theoretically predicted asymmetric $m$ vs $n$ trade-off.
