Generalization of the Kimeldorf-Wahba correspondence for constrained interpolation

TL;DR

This paper extends the classical connection between Bayesian estimation and optimal interpolation in RKHS to include linear inequality constraints, showing that the MAP estimator, not the mean, provides the optimal constrained interpolation.

Contribution

It generalizes the Kimeldorf-Wahba correspondence to constrained cases, establishing that the MAP estimator is the optimal solution under inequality constraints.

Findings

MAP estimator is the optimal constrained interpolation function.

The correspondence holds with the MAP, not the mean, of the posterior.

Numerical example confirms theoretical results.

Abstract

In this paper, we extend the correspondence between Bayes' estimation and optimal interpolation in a Reproducing Kernel Hilbert Space (RKHS) to the case of linear inequality 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.