Constrained Optimal Smoothing and Bayesian Estimation

TL;DR

This paper extends the link between Bayesian estimation and optimal smoothing in RKHS by incorporating convex constraints, proving that the MAP estimate corresponds to the constrained smoothing function.

Contribution

It generalizes previous work by establishing the MAP as the optimal constrained smoothing function in RKHS with convex constraints.

Findings

MAP equals the optimal constrained smoothing function in RKHS

The approach uses approximating Hilbertian spaces and discretized models

Generalizes Kimeldorf-Wahba's result to constrained cases

Abstract

In this paper, we extend the correspondence between Bayesian estimation and optimal smoothing in a Reproducing Kernel Hilbert Space (RKHS) adding a convexe constraints 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 [7] of Kimeldorf-Wahba where it is proved that the optimal smoothing solution is the mean of the posterior distribution.