  Parameter estimation, statistical tests and confidence sets are the
cornerstones of classical statistics that allow scientists to make inferences
about the underlying process that generated the observed data. A key question
is whether one can still construct hypothesis tests and confidence sets with
proper coverage and high power in a so-called likelihood-free inference (LFI)
setting; that is, a setting where the likelihood is not explicitly known but
one can forward-simulate observable data according to a stochastic model. In
this paper, we present $\texttt{ACORE}$ (Approximate Computation via Odds Ratio
Estimation), a frequentist approach to LFI that first formulates the classical
likelihood ratio test (LRT) as a parametrized classification problem, and then
uses the equivalence of tests and confidence sets to build confidence regions
for parameters of interest. We also present a goodness-of-fit procedure for
checking whether the constructed tests and confidence regions are valid.
$\texttt{ACORE}$ is based on the key observation that the LRT statistic, the
rejection probability of the test, and the coverage of the confidence set are
conditional distribution functions which often vary smoothly as a function of
the parameters of interest. Hence, instead of relying solely on samples
simulated at fixed parameter settings (as is the convention in standard Monte
Carlo solutions), one can leverage machine learning tools and data simulated in
the neighborhood of a parameter to improve estimates of quantities of interest.
We demonstrate the efficacy of $\texttt{ACORE}$ with both theoretical and
empirical results. Our implementation is available on Github.
