Probabilistic learning constrained by realizations using a weak formulation of Fourier transform of probability measures

This paper deals with the taking into account a given target set of realizations as constraints in Kullback–Leibler divergence minimum principle (KLDMP). We present novel probabilistic learning algorithm that makes it possible to use KLDMP when are not defined by statistical moments for quantity interest (QoI) an uncertain/stochastic computational model, but QoI which associated these or cannot be estimated. The method consists defining functional constraint, equality Fourier transforms posterior probability measure and measure, constructing finite representation weak formulation this constraint. proposed approach allows estimating (unsupervised case) joint control parameter (supervised case). existence uniqueness is analyzed two cases. numerical aspects detailed order facilitate implementation method. presented application high dimension demonstrates efficiency robustness algorithm.