Natesh S Pillai
Associate ProfessorHarvard University
Levy Random Measures: Posterior Consistency and Applications
Non-parametric function estimation using Lévy random measures is a very active area of current research. In this thesis further contributions, both theoretical and methodological, are made towards non-parametric function estimation using Lévy random measures. In chapter 2, it is observed that Lévy random measures lead to a unified perspective of non-parametric function estimation using Bayesian methods and those using kernel methods such as Tikhonov regularization used in the machine learning literature. A coherent Bayesian kernel model based on an integral operator defined as the convolution of a kernel with a signed measure is studied. A few results on Fredholm integral operators are derived and a general class of measures whose image is dense in the reproducing kernel Hilbert space (RKHS) induced by the kernel is identified. These results lead to a function theoretic foundation for using non-parametric prior specifications in Bayesian modeling, such as Gaussian process and Dirichlet process prior distributions. In chapter 3, easily verifiable conditions are derived for posterior consistency to hold in commonly used regression models with prior distributions on infinite dimensional spaces constructed from Lévy random fields. On route to proving conistency, convergence properties of finite dimensional approximations of Lévy random fields are studied. The key technical issues involved are outlined, and the results are illustrated by proving the posterior consisteny in concrete examples. In chapter 4, the posterior consistency for non-parametric Poisson regression models is proved. The key step is to construct test functions that separate points, and have exponentially decaying type I and II errors. In chapter 5, a novel application of Lévy random measures is discussed. It is shown that Lévy random measures can be used for constructing prior distributions for spectral measures. Together with Bochner�s theorem, this leads to a construction of non-parametric prior distributions on the cone of positive definite functions.