This paper tackles the challenge presented by small-data to the task of Bayesian inference. A novel methodology, based on manifold learning and manifold sampling, is proposed for solving this computational statistics problem under the following assumptions: (1) neither the prior model nor the likelihood function are Gaussian and neither can be approximated by a Gaussian measure; (2) the number of functional input (system parameters) and functional output (quantity of interest) can be large; (3) the number of available realizations of the prior model is small, leading to the small-data challenge typically associated with expensive numerical simulations; the number of experimental realizations is also small; (4) the number of the posterior realizations required for decision is much larger than the available initial dataset. The method and its mathematical aspects are detailed. Three applications are presented for validation: The first two involve mathematical constructions aimed to develop intuition around the method and to explore its performance. The third example aims to demonstrate the operational value of the method using a more complex application related to the statistical inverse identification of the non-Gaussian matrix-valued random elasticity field of a damaged biological tissue (osteoporosis in a cortical bone) using ultrasonic waves.

中文翻译：

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利用来自小数据集的流形上的非高斯概率学习对贝叶斯后验进行采样

本文解决了小数据对贝叶斯推理任务提出的挑战。在以下假设的基础上，提出了一种基于流形学习和流形采样的新颖方法来解决该计算统计问题：（1）先验模型和似然函数都不是高斯函数，也不可以通过高斯测度近似。（2）功能输入（系统参数）和功能输出（感兴趣的数量）的数量可能很大；（3）现有模型的可用实现数量很少，从而导致通常与昂贵的数值模拟相关的小数据挑战；实验实现的数量也很少；（4）决策所需的后验实现数量远大于可用的初始数据集。详细介绍了该方法及其数学方面。提出了三个用于验证的应用程序：前两个应用程序涉及数学构造，旨在发展围绕该方法的直觉并探索其性能。第三个示例旨在展示使用更复杂的应用的方法的操作价值，该应用与使用超声波对受损生物组织（皮质骨中的骨质疏松症）的非高斯矩阵值随机弹性场进行统计逆识别有关。