Optimal Smoothing and Gaussian Processes with noisy data under constraints

Laurence Grammont (1, 2) , Xavier Bay (3, 4, 5, 6) , Hassan Maatouk (7)

1 ICJ - Institut Camille Jordan
2 MMCS - Modélisation mathématique, calcul scientifique
3 Mines Saint-Étienne MSE - École des Mines de Saint-Étienne
4 LIMOS - Laboratoire d'Informatique, de Modélisation et d'Optimisation des Systèmes
5 FAYOL-ENSMSE - Institut Henri Fayol
6 FAYOL-ENSMSE - Département Génie mathématique et industriel
7 AGM - UMR 8088 - Analyse, Géométrie et Modélisation

In this paper, we extend the correspondence between Bayesian estimation and optimal smoothing in a Reproducing Kernel Hilbert Space (RKHS) adding a convex constraint on the solution. Through a sequence of approximating Hilbertian spaces and a discretized model, we prove that the Maximum A Posteriori (MAP) of the posterior distribution is exactly the optimal constrained smoothing function in the RKHS. This paper can be read as a generalization of the paper [11] of Kimeldorf-Wahba where it is proved that the optimal smoothing solution is the mean of the posterior distribution. This is also a generalization of the paper [4] where the case of constrained optimal interpolation is treated. Here we relax the interpolation by introducing noise effect in the data. A numerical example is given to illustrate the theoretical result of this paper.

Domaines: Méthodologie [stat.ME]; Mathématiques [math]; Mathématiques générales [math.GM]; Probabilités [math.PR]

Laurence Grammont, Xavier Bay, Hassan Maatouk. Optimal Smoothing and Gaussian Processes with noisy data under constraints. 2022. ⟨hal-03625227⟩