The exact relationship between regularization in RKHS and Bayesian estimation of Gaussian random fields

Reconstruction of a function from noisy data is often formulated as a regularized optimization problem over an infinite-dimensional reproducing kernel Hilbert space (RKHS). The solution describes the observed data and also has a small RKHS norm. When the data fit is measured using a quadratic loss, this estimator has a known statistical interpretation in terms of Gaussian random fields: it provides the minimum variance estimate of the unknown function given the noisy measurements. In this paper, we provide a statistical interpretation when more general losses are used, such as Vapnik or Huber. For a given data set, the RKHS function estimate contains all the possible maximum a posteriori estimates of a random field for which the prior distribution is Gaussian.