Generalization of the Kimeldorf-Wahba correspondence for constrained interpolation

Xavier Bay (1) , Laurence Grammont (2, 3) , Hassan Maatouk (4)

1 UR-LSTI-ENSMSE - Laboratoire en Sciences et Technologies de l'Information
2 ICJ - Institut Camille Jordan
3 MMCS - Modélisation mathématique, calcul scientifique
4 AGM - UMR 8088 - Analyse, Géométrie et Modélisation

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.

Mots clés: Correspondence; interpolation; inequality constraints; reproducing Kernel Hilbert space; Gaussian process; Bayesian estimation; maximum a posteriori

Domaines: Statistiques [math.ST]

Xavier Bay, Laurence Grammont, Hassan Maatouk. Generalization of the Kimeldorf-Wahba correspondence for constrained interpolation. Electronic Journal of Statistics , 2016, 10 (1), pp.1580-1595. ⟨10.1214/16-ejs1149⟩. ⟨hal-04150103⟩