A Nonparametric Conjugate Prior Distribution for the Maximizing Argument of a Noisy Function[PDF] [BibTeX]
We propose a novel Bayesian approach to solve stochastic optimization problems that involve ﬁnding extrema of noisy, nonlinear functions. Previous work has focused on representing possible functions explicitly, which leads to a two-step procedure of ﬁrst, doing inference over the function space and second, ﬁnding the extrema of these functions. Here we skip the representation step and directly model the distribution over extrema. To this end, we devise a non-parametric conjugate prior where the natural parameter corresponds to a given kernel function and the sufﬁcient statistic is composed of the observed function values. The resulting posterior distribution directly captures the uncertainty over the maximum of the unknown function.