Advanced Mathematical Statistics and Probability Theory

This course reviews concepts and tools of advanced mathematical analyses of statistical methods. Topics include empirical process theory, minimax theory and concentration of measures for high dimensional problems, random matrix theory, sparse estimation theory, asymptotic theory of nonparametric Bayesian methods, empirical Bayesian theory, stochastic process modeling, diffusion process theory, optimization, etc. The course emphasizes on the duality of Bayesian and frequentist approaches and explores benefits arising from their synthesis. It will introduce advanced topics in probability theory, functional analysis, and topology as needed.