Topics in kernel hypothesis testing

This thesis investigates some unaddressed problems in kernel nonparametrichypothesis testing. The contributions are grouped around three main themes:Wild Bootstrap for Degenerate Kernel Tests. A wild bootstrap method for non-parametric hypothesis tests based on kernel distribution embeddings is pro-posed. This bootstrap method is used to construct provably consistent teststhat apply to random processes. It applies to a large group of kernel testsbased on V-statistics, which are degenerate under the null hypothesis, andnon-degenerate elsewhere. In experiments, the wild bootstrap gives strongperformance on synthetic examples, on audio data, and in performance bench-marking for the Gibbs sampler.A Kernel Test of Goodness of Fit. A nonparametric statistical test for goodness-of-fit is proposed: given a set of samples, the test determines how likely itis that these were generated from a target density function. The measure ofgoodness-of-fit is a divergence constructed via Stein’s method using functionsfrom a Reproducing Kernel Hilbert Space. Construction of the test is based onthe wild bootstrap method. We apply our test to quantifying convergence ofapproximate Markov Chain Monte Carlo methods, statistical model criticism,and evaluating quality of fit vs model complexity in nonparametric densityestimation.Fast Analytic Functions Based Two Sample Test. A class of nonparametric two-sample tests with a cost linear in the sample size is proposed. Two tests aregiven, both based on an ensemble of distances between analytic functionsrepresenting each of the distributions. Experiments on artificial benchmarksand on challenging real-world testing problems demonstrate good power/timetradeoff retained even in high dimensional problems.