Posterior consistency in nonparametric regression problems under Gaussian process prior

In the Bayesian approach to statistical inference, the posterior distribution summarizes information regarding unknown parameters after observing data. As the sample size increases, we expect that the posterior distribution will concentrate around the true value of the parameter. This is known as posterior consistency. Posterior consistency is sometimes used as a theoretical justification of the Bayesian method.;In this dissertation, we establish posterior consistency in nonparametric regression problems using Gaussian process priors, known as Gaussian process regression. Gaussian process regression is one of the most popular Bayesian nonparametric regression approaches, and it consists of nonparametrically modeling the unknown regression function as a Gaussian process a priori.;We provide an extension of the posterior consistency theorem of Schwartz (1965) for independent but non-identically distributed observations. Then we apply our theorem to the Gaussian process regression problem by giving conditions on the Gaussian process prior that allow us to verify the conditions of the consistency theorem, namely the prior positivity of neighborhoods of the true parameter and the existence of tests with suitable properties.;The specific nonparametric regression problem that we consider assumes that the data have normal or Laplace (double exponential) noise distributions. The error variance is also assumed to be unknown and needs to be estimated. We consider three different metric topologies on the space of regression functions, and then define joint neighborhoods of the regression function and the noise variance. The joint neighborhoods are based on the L1 metric, an in-probability metric, and the Hellinger metric. For each of these joint neighborhoods, we prove almost sure consistency of the posterior distribution by showing that the posterior probability of these joint neighborhoods converges almost surely to 1 under appropriate conditions on the regression function and covariates. Specifically, when the covariate values are fixed in advance, we prove almost sure consistency based on the L1 metric and the in probability metric assuming the covariate values are spread out sufficiently well. When the covariate values are sampled from a probability distribution, we prove almost sure consistency based on the in-probability metric and the Hellinger metric. Furthermore, in the random covariate case, under an additional uniform boundedness condition, we prove almost sure consistency with the L1 metric as well.;Finally, we also study how much posterior consistency can be extended into other regression model structures, which deal with hyperparameters of a covariance function and multidimensional covariates.