Nonparametric Bayesian Regression

Abstract

Summary: ``It is desired to estimate a real-valued function $F$ on the unit square having observed $F$ with error at $N$ points in the square. $F$ is assumed to be drawn from a particular Gaussian process and measured with independent Gaussian errors. The proposed estimate is the Bayes estimate of $F$ given the data. The roughness penalty corresponding to the prior is derived and it is shown how the Bayesian technique can be regarded as a generalisation of variance components analysis. The proposed estimate is shown to be consistent in the sense that the expected squared error averaged over the data points converges to zero as $N\to\infty$. Upper bounds on the order of magnitude of the expected average squared error are calculated. The proposed technique is compared with existing spline techniques in a simulation study. Generalisations to higher dimensions are discussed.''