{"id":{"repo_id":"mit","oai_identifier":"oai:dspace.mit.edu:1721.1/155358"},"canonical_url":"https://search.dev.ndltd.org/etd/mit/oai:dspace.mit.edu:1721.1/155358","repository":{"repo_id":"mit","name":"MIT","base_url":"https://dspace.mit.edu/oai/request"},"display":{"title":"Likelihood-Free Hypothesis Testing and Applications of the Energy Distance","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_html":"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_has_math":false,"creators":["Gerber, Patrik Róbert"],"institution":"Massachusetts Institute of Technology","degree_name":"Doctoral","degree_level":null,"degree_discipline":null,"degree_department":"Massachusetts Institute of Technology. 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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."]},{"key":"dc:description.degree","label":"Dc Description Degree","values":["Ph.D."]},{"key":"dc:title","label":"Title","values":["Likelihood-Free Hypothesis Testing and Applications of the Energy Distance"]}]}],"canonical_facts":{"dc:contributor.advisor":["Rigollet, Philippe"],"dc:contributor.department":["Massachusetts Institute of Technology. Department of Mathematics"],"dc:creator":["Gerber, Patrik Róbert"],"dc:date.accessioned":["2024-06-27T19:47:32Z"],"dc:date.available":["2024-06-27T19:47:32Z"],"dc:date.issued":["2024-05"],"dc:description.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. 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