A methodology to estimate from samples the probability density of a random variable x conditional to the values of a set of covariates $$\{z_{l}\}$$ { z l } is proposed. The methodology relies on a data-driven formulation of the Wasserstein barycenter, posed as a minimax problem in terms of the conditional map carrying each sample point to the barycenter and a potential characterizing the inverse of this map. This minimax problem is solved through the alternation of a flow developing the map in time and the maximization of the potential through an alternate projection procedure. The dependence on the covariates $$\{z_{l}\}$$ { z l } is formulated in terms of convex combinations, so that it can be applied to variables of nearly any type, including real, categorical and distributional. The methodology is illustrated through numerical examples on synthetic and real data. The real-world example chosen is meteorological, forecasting the temperature distribution at a given location as a function of time, and estimating the joint distribution at a location of the highest and lowest daily temperatures as a function of the date.

中文翻译：

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通过最优传输进行条件密度估计和模拟

提出了一种从样本中估计随机变量 x 的概率密度的方法，该概率密度取决于一组协变量 $$\{z_{l}\}$$ { zl } 的值。该方法依赖于 Wasserstein 重心的数据驱动公式，根据将每个样本点带到重心的条件映射和表征该映射逆的潜在特征，提出了一个极小极大问题。这个极小极大问题是通过交替绘制地图的流和通过交替投影程序最大化潜力来解决的。对协变量 $$\{z_{l}\}$$ { zl } 的依赖是根据凸组合来表达的，因此它可以应用于几乎任何类型的变量，包括实数、分类和分布。该方法通过合成数据和真实数据的数值例子来说明。选择的实际示例是气象学，预测给定位置的温度分布作为时间的函数，并估计每日最高和最低温度位置的联合分布作为日期的函数。