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# Confidence Sets and Hypothesis Testing in a Likelihood-Free Inference Setting

*Proceedings of the 37th International Conference on Machine Learning*, PMLR 119:2323-2334, 2020.

#### Abstract

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 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. 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 ACORE with both theoretical and empirical results. Our implementation is available on Github.