Chapter 8 - Bayesian spectral analysis regression

Abstract

Bayesian nonparametric regression requires prior distributions on the space of smooth functions. Gaussian process priors provide a natural model for the nonparametric smooth functions. The mean of the Gaussian process is the prior mean; the variance controls the trade-off between the prior mean and data; and the correlation determines the smoothness of the sample paths. This chapter presents the spectral analysis of Gaussian processes. Spectral analysis decomposes the Gaussian process into a lower-dimensional, linear model that is simpler to estimate. Hierarchical Bayes priors on the spectral coefficients determine the covariance structure of the Gaussian process and the amount of smoothing. The Bayesian analysis estimates the hyperparameters for the smoothing prior and avoids cross-validation. The spectral analysis facilitates extending the model to include shape constraints and nonnormal likelihoods. The chapter illustrates estimating these models with the R library bsamGP.