Bayesian analysis of nonparametric regression problems

Messan Guodjo Amewou-Atisso, Purdue University

Abstract

We consider Bayesian inference in the linear regression problem with an unknown error distribution that is symmetric around zero. We show that if the prior for the error distribution assigns positive probabilities to Kullback-Leibler neighborhoods of the true distribution, then the posterior distribution is consistent in the weak topology. This, in particular, implies that the posterior distributions of the regression parameters are consistent in the Euclidean Metric. The result follows from our generalization of a celebrated result of Schwartz to the case of independent, non-identically distributed random variables. We then specialize to two important families of prior distributions, namely the Polya tree and Dirichlet mixtures and show that under appropriate conditions, these priors satisfy the positivity requirement of the prior probabilities of the neighborhoods of the true density. An important auxiliary result, needed for application of Schwartz's theorem is the existence of exponentially consistent tests. We consider the case of both non-stochastic and stochastic regressors. Extensive simulations have been performed to get more insight into the performance of the Dirichlet mixtures on the posterior of the slope parameter. Similar problems of Bayesian inference in random effects models and in a generalized linear model for binary responses with unknown link function are also considered. Two examples are given for the Bioassay problem.

Degree

Ph.D.

Advisors

Ghosh, Purdue University.

Subject Area

Statistics

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