Contributions to Bayesian nonparametrics and Bayesian robustness

Nariankadu Datatreya Shyamalkumar, Purdue University

Abstract

The first part of the thesis concerns itself with Bayesian nonparametrics. We consider the problem of estimating a survival function; the survival function is known to have a form of the type $(1-F\sb0)\cdot (1-G),$ with $F\sb0$ known. It is known in the literature that the generalized maximum likelihood estimator is inconsistent when one is working with continuous distributions. We consider the Bayesian approach to this problem; working with the Dirichlet process prior we show consistency in the discrete case and inconsistency in the continuous case. For the continuous case we use a prior which concentrates its mass on the set of absolutely continuous distributions by mixing a uniform distribution with a distribution drawn from a Dirichlet. In the latter case we exhibit consistency for a large class of distributions. Computation of the estimator so derived is also discussed using an importance sampling scheme. In the second part of the thesis we deal with cyclic Kullback-Leibler projections. We first study the convergence of iterations of a certain operator on the space of probability measures; an example of which turns out to be the EM algorithm for estimating finite mixtures. There is some overlap with results in the literature. In the second problem we show how this method could be used to prove convergence of the Data Augmentation method. In the third part of the thesis we consider two problems in Bayesian Robustness. We first deal with a problem in likelihood robustness. We show that if the likelihood can be embedded in a class with a special convex structure then we can find approximations to bounds of posterior expectations of functions of interest. In the second problem we give conditions on a class of priors which ensure that the range of posterior expectation of a bounded function of interest, as the prior varies over the class, converges to a point. Examples of classes which satisfy this condition are given.

Degree

Ph.D.

Advisors

Berger, Purdue University.

Subject Area

Statistics

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