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Rigollet, Philippe Gerber, Patrik Róbert Massachusetts Institute of Technology. Department of Mathematics 2024-06-27T19:47:32Z 2024-06-27T19:47:32Z 2024-05 2024-05-15T16:20:13.108Z https://hdl.handle.net/1721.1/155358 This thesis studies questions in nonparametric testing and estimation that are inspired by machine learning. One of the main problems of our interest is likelihood-free hypothesis testing: given three samples X, Y and Z with sample sizes n, n and m respectively, one must decide whether the distribution of Z is closer to that of X or that of Y . We fully characterize the problem’s sample complexity for multiple distribution classes and with high probability. We uncover connections to two-sample, goodness-of-fit and robust testing, and show the existence of a trade-off of the form mn ≍ k/ε^4, where k is an appropriate notion of complexity and ε is the total variation separation between the distributions of X and Y . We generalize our problem to allow Z to come from a mixture of the distributions of X and Y , and propose a kernel-based test for its solution, and also verify the existence of a trade-off between m and n on experimental data from particle physics. In addition, we demonstrate that the family of “classifier accuracy” tests are not only popular in practice but also provably near-optimal, recovering and simplifying a multitude of classical and recent results. Finally, we study affine classifiers as a tool for estimation and testing, with the key technical tool being a connection to the energy distance. In particular, we propose a density estimation routine based on minimizing the generalized energy distance, targeting smooth densities and Gaussian mixtures. We interpret our results in terms of half-space separability over these classes, and derive analogous results for discrete distributions. As a consequence we deduce that any two discrete distributions are well-separated by a half-space, provided their support is embedded as a packing of a high-dimensional unit ball. We also scrutinize two recent applications of the energy distance in the two-sample testing literature. Ph.D. Massachusetts Institute of Technology In Copyright - Educational Use Permitted Copyright retained by author(s) https://rightsstatements.org/page/InC-EDU/1.0/ Likelihood-Free Hypothesis Testing and Applications of the Energy Distance Thesis application/pdf Doctoral Doctor of Philosophy Publication unknown unknown <Publication xmlns="https://www.openaire.eu/cerif-profile/1.1/" id="ae2fa008-45f3-4970-b706-ca40ffa3411f"> <Type xmlns="https://www.openaire.eu/cerif-profile/vocab/COAR_Publication_Types">http://purl.org/coar/resource_type/c_1843</Type> <Title>Likelihood-Free Hypothesis Testing and Applications of the Energy Distance</Title> <PublishedIn> <Publication> </Publication> </PublishedIn> <PublicationDate>2024-05</PublicationDate> <Authors> <Author> <DisplayName>Gerber, Patrik Róbert</DisplayName> <Affiliation> <OrgUnit> </OrgUnit> </Affiliation> </Author> </Authors> <Editors> </Editors> <Publishers> <Publisher> <DisplayName>Massachusetts Institute of Technology</DisplayName> <OrgUnit /> </Publisher> </Publishers> <License>https://rightsstatements.org/page/InC-EDU/1.0/</License> <Abstract>This thesis studies questions in nonparametric testing and estimation that are inspired by machine learning. One of the main problems of our interest is likelihood-free hypothesis testing: given three samples X, Y and Z with sample sizes n, n and m respectively, one must decide whether the distribution of Z is closer to that of X or that of Y . We fully characterize the problem’s sample complexity for multiple distribution classes and with high probability. We uncover connections to two-sample, goodness-of-fit and robust testing, and show the existence of a trade-off of the form mn ≍ k/ε^4, where k is an appropriate notion of complexity and ε is the total variation separation between the distributions of X and Y . We generalize our problem to allow Z to come from a mixture of the distributions of X and Y , and propose a kernel-based test for its solution, and also verify the existence of a trade-off between m and n on experimental data from particle physics. In addition, we demonstrate that the family of “classifier accuracy” tests are not only popular in practice but also provably near-optimal, recovering and simplifying a multitude of classical and recent results. Finally, we study affine classifiers as a tool for estimation and testing, with the key technical tool being a connection to the energy distance. In particular, we propose a density estimation routine based on minimizing the generalized energy distance, targeting smooth densities and Gaussian mixtures. We interpret our results in terms of half-space separability over these classes, and derive analogous results for discrete distributions. As a consequence we deduce that any two discrete distributions are well-separated by a half-space, provided their support is embedded as a packing of a high-dimensional unit ball. We also scrutinize two recent applications of the energy distance in the two-sample testing literature.</Abstract> <Access xmlns="http://purl.org/coar/access_right" > </Access> </Publication>