• Course Overview: This course is about Advanced Bayesian Statistics. We focus on topics beyond the usual intro level Bayesian class, including Bayesian variable selections, the mixture of experts, Gaussian process, and large sample properties.

• Notes This is a very preliminary note.

• Table of Contents

  1. Bayesian Variable Selection
     • Zellner's g-priors
     • Spike and slab priors
     • Laplace approximation
     • Predictive density

  2. The EM
     • The algorithm
     • The math behind EM
     • Examples in linear regressions

  3. Gaussian Mixture Models
     • Mixing densities
     • Mixing regressions
     • Mixture of experts

  4. Gaussian Process
     • Reproducing kernel Hilbert space and Mercer's theorem
     • Predictive density
     • Asymptotics of the predictive density

  5. Objective Priors
     • Shannon's information theory
     • Jeffreys' prior
     • Reference prior

  6. Posterior Large Sample Properties
     • Contraction
     • Bernstein von Mises theorem
     • MCMC confidence interval