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This paper deals with the taking into account a given set of realizations asconstraints in the Kullback-Leibler minimum principle, which is used as aprobabilistic learning algorithm. This permits the effective integration ofdata into predictive models. We consider the probabilistic learning of a randomvector that is made up of either a quantity of interest (unsupervised case) orthe couple of the quantity of interest and a control parameter (supervisedcase). A training set of independent realizations of this random vector isassumed to be given and to be generated with a prior probability measure thatis unknown. A target set of realizations of the QoI is available for the twoconsidered cases. The framework is the one of non-Gaussian problems in highdimension. A functional approach is developed on the basis of a weakformulation of the Fourier transform of probability measures (characteristicfunctions). The construction makes it possible to take into account the targetset of realizations of the QoI in the Kullback-Leibler minimum principle. Theproposed approach allows for estimating the posterior probability measure ofthe QoI (unsupervised case) or of the posterior joint probability measure ofthe QoI with the control parameter (supervised case). The existence and theuniqueness of the posterior probability measure is analyzed for the two cases.The numerical aspects are detailed in order to facilitate the implementation ofthe proposed method. The presented application in high dimension demonstratesthe efficiency and the robustness of the proposed algorithm.

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