标题： 基于实现实的概率学习，使用概率测度傅里叶变换的弱形式 显示英文标题

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

摘要： 本文讨论了在Kullback-Leibler最小原则中考虑给定的一组实现作为约束，该原则被用作概率学习算法。这使得数据能够有效地整合到预测模型中。我们考虑了一个随机向量的概率学习，该随机向量由感兴趣的量（无监督情况）或感兴趣的量和控制参数的组合（监督情况）组成。假设给出了一组独立的该随机向量的实现，并且这些实现是通过一个未知的先验概率测度生成的。对于两种情况，都提供了感兴趣的量（QoI）的实现目标集。框架是高维非高斯问题。基于概率测度（特征函数）的傅里叶变换的弱公式，发展了一种函数方法。该构造使得能够在Kullback-Leibler最小原则中考虑QoI的实现目标集。所提出的方法允许估计QoI的后验概率测度（无监督情况）或QoI与控制参数的后验联合概率测度（监督情况）。分析了两种情况下后验概率测度的存在性和唯一性。详细描述了数值方面，以促进所提出方法的实现。所展示的高维应用证明了所提出算法的效率和鲁棒性。 显示英文摘要

摘要： This paper deals with the taking into account a given set of realizations as constraints in the Kullback-Leibler minimum principle, which is used as a probabilistic learning algorithm. This permits the effective integration of data into predictive models. We consider the probabilistic learning of a random vector that is made up of either a quantity of interest (unsupervised case) or the couple of the quantity of interest and a control parameter (supervised case). A training set of independent realizations of this random vector is assumed to be given and to be generated with a prior probability measure that is unknown. A target set of realizations of the QoI is available for the two considered cases. The framework is the one of non-Gaussian problems in high dimension. A functional approach is developed on the basis of a weak formulation of the Fourier transform of probability measures (characteristic functions). The construction makes it possible to take into account the target set of realizations of the QoI in the Kullback-Leibler minimum principle. The proposed approach allows for estimating the posterior probability measure of the QoI (unsupervised case) or of the posterior joint probability measure of the QoI with the control parameter (supervised case). The existence and the uniqueness of the posterior probability measure is analyzed for the two cases. The numerical aspects are detailed in order to facilitate the implementation of the proposed method. The presented application in high dimension demonstrates the efficiency and the robustness of the proposed algorithm.