The Annals of Statistics

Nonparametric Bayesian Regression

Daniel Barry

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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 $N\rightarrow\infty$. Upper bounds on the order of magnitude 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.

Article information

Ann. Statist., Volume 14, Number 3 (1986), 934-953.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62G05: Estimation
Secondary: 62J05: Linear regression 62M99: None of the above, but in this section

Bayes estimate Brownian sheet roughness penalty consistency


Barry, Daniel. Nonparametric Bayesian Regression. Ann. Statist. 14 (1986), no. 3, 934--953. doi:10.1214/aos/1176350043.

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