Carnegie Mellon University
Browse

Posterior Consistency in Nonparametric Regression Problems under Gaussian Process Priors

Download (309.35 kB)
journal contribution
posted on 2007-04-01, 00:00 authored by Taeryon Choi, Mark J. Schervish

Posterior consistency can be thought of as a theoretical justification of the Bayesian method. One of the most popular approaches to nonparametric Bayesian regression is to put a nonparametric prior distribution on the unknown regression function using Gaussian processes. In this paper, we study posterior consistency in nonparametric regression problems using Gaussian process priors. We use an extension of the theorem of Schwartz (1965) for nonidentically distributed observations, verifying its conditions when using Gaussian process priors for the regression function with normal or double exponential (Laplace) error distributions. We define a metric topology on the space of regression functions and then establish almost sure consistency of the posterior distribution. Our metric topology is weaker than the popular L1 topology. With additional assumptions, we prove almost sure consistency when the regression functions have L1 topologies. When the covariate (predictor) is assumed to be a random variable, we prove almost sure consistency for the joint density function of the response and predictor using the Hellinger metrics.

History

Publisher Statement

All Rights Reserved

Date

2007-04-01

Usage metrics

    Exports

    RefWorks
    BibTeX
    Ref. manager
    Endnote
    DataCite
    NLM
    DC