Electronic Journal of Statistics

Generalization of the Kimeldorf-Wahba correspondence for constrained interpolation

Xavier Bay, Laurence Grammont, and Hassan Maatouk

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In this paper, we extend the correspondence between Bayes’ estimation and optimal interpolation in a Reproducing Kernel Hilbert Space (RKHS) to the case of convex constraints such as boundedness, monotonicity or convexity. In the unconstrained interpolation case, the mean of the posterior distribution of a Gaussian Process (GP) given data interpolation is known to be the optimal interpolation function minimizing the norm in the RKHS associated to the GP. In the constrained case, we prove that the Maximum A Posteriori (MAP) or Mode of the posterior distribution is the optimal constrained interpolation function in the RKHS. So, the general correspondence is achieved with the MAP estimator and not the mean of the posterior distribution. A numerical example is given to illustrate this last result.

Article information

Electron. J. Statist., Volume 10, Number 1 (2016), 1580-1595.

Received: February 2016
First available in Project Euclid: 31 May 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G15: Gaussian processes 60G25: Prediction theory [See also 62M20] 62K20: Response surface designs

Correspondence interpolation inequality constraints reproducing Kernel Hilbert space Gaussian process Bayesian estimation maximum a posteriori


Bay, Xavier; Grammont, Laurence; Maatouk, Hassan. Generalization of the Kimeldorf-Wahba correspondence for constrained interpolation. Electron. J. Statist. 10 (2016), no. 1, 1580--1595. doi:10.1214/16-EJS1149. https://projecteuclid.org/euclid.ejs/1464710242

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