Thesis Proposal: Non-parametric Hyper Markov Priors

Markov distributions are used to describe multivariate data with conditional independence structure. Applications of Markov distributions arise in many fields including demography, flood prediction, and telecommunications. A hyper Markov law is a distribution over the space of all Markov distributions; such laws have been used as prior distributions for various types of graphical models. Dirichlet processes have also been used to specify priors in a non-parametric form. I have developed a family of non-parametric hyper Markov laws that I call hyper Dirichlet processes, which combine the separate ideas of hyper Markov laws and non-parametric prior processes. In my thesis, I propose to describe these distributions and their properties, and to apply them to specific problems. For example, I define a hyper Markov mixture of Gaussians and use it in the form of a hyper Markov prior to provide a non-parametric way to mix graphical Gaussian distributions.