Statistical Theory and Methodology

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The investigators consider a class of problems in probability and statistical inference arising from large-scale scientific investigations. Massive hypothesis testing situations, with tens or hundreds of thousands of cases to consider simultaneously, are treated using empirical Bayes methods, as an efficient compromise between frequentist and Bayesian analyses. In a microarray experiment, for example, an empirical Bayes False Discovery Rate approach helps sift through thousands of z-values when searching for those genes of significant interest. Special attention if given to the appropriate choice of the null hypothesis, which may be quite different than classical text-book recipes. Markov chain monte carlo tech- niques are analyzed using eigen decompositions of the transition matrix to quantify how quickly initial conditions dissipate as the sampling algorithm proceeds toward steady state. Advanced applications involve implementing random sampling over curved manifolds in high-dimensional spaces.The general theme of this project is the development of mathematical methods of genuine practical utility, both in probability and statistical inference. Classical statistical methods were developed in a scientific world of small data sets investigating individual questions. Modern scientific technology, like microarrays, fMRI devices, and satellite imagers, now produce enormous data sets that aim to simultaneously investigate thousands of possibilities. This project proposes methods for dealing with the new scientific environment. New techniques of statistical inference and probabilistic modelling are proposed, based on recent experience in the biological, social science, and physical science worlds.