Confidence Sets and Hypothesis Testing in a Likelihood-Free Inference Setting
Niccolo Dalmasso,u00a0Rafael Izbicki,u00a0Ann Lee
Parameter estimation, statistical tests and conufb01dence 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 conufb01dence 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 ACORE (Approximate Computation via Odds Ratio Estimation), a frequentist approach to LFI that ufb01rst formulates the classical likelihood ratio test (LRT) as a parametrized classiufb01cation problem, and then uses the equivalence of tests and conufb01dence sets to build conufb01dence regions for parameters of interest. We also present a goodness-of-ufb01t procedure for checking whether the constructed tests and conufb01dence regions are valid. ACORE is based on the key observation that the LRT statistic, the rejection probability of the test, and the coverage of the conufb01dence 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 ufb01xed 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 efufb01cacy of ACORE with both theoretical and empirical results. Our implementation is available on Github.
