Optimal Smoothing and Gaussian Processes with noisy data under constraints

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