Bayesian covariance estimation and inference in latent Gaussian process models

Abstract

This paper describes inference methods for functional data under the assumption that the functional data of interest are smooth latent functions, characterized by a Gaussian process, which have been observed with noise over a finite set of time points. The methods we propose are completely specified in a Bayesian environment that allows for all inferences to be performed through a simple Gibbs sampler. Our main focus is in estimating and describing uncertainty in the covariance function. However, these models also encompass functional data estimation, functional regression where the predictors are latent functions, and an automatic approach to smoothing parameter selection. Furthermore, these models require minimal assumptions on the data structure as the time points for observations do not need to be equally spaced, the number and placement of observations are allowed to vary among functions, and special treatment is not required when the number of functional observations is less than the dimensionality of those observations. We illustrate the effectiveness of these models in estimating latent functional data, capturing variation in the functional covariance estimate, and in selecting appropriate smoothing parameters in both a simulation study and a regression analysis of medfly fertility data.