Stochastic inversion problems are typically encountered when it is wanted to quantify the uncertainty affecting the inputs of computer models. They consist in estimating input distributions from noisy, observable outputs, and such problems are increasingly examined in Bayesian contexts where the targeted inputs are affected by stochastic uncertainties. In this regard, a stochastic input can be qualified as meaningful if it explains most of the output uncertainty. While such inverse problems are characterized by identifiability conditions, constraints of "signal to noise", that can formalize this meaningfulness, should be accounted for within the definition of the model, prior to inference. This article investigates the possibility of forcing a solution to be meaningful in the context of parametric uncertainty quantification, through the tools of global sensitivity analysis and information theory (variance, entropy, Fisher information). Such forcings have mainly the nature of constraints placed on the input covariance, and can be made explicit by considering linear or linearizable models. Simulated experiments indicate that, when injected into the modeling process, these constraints can limit the influence of measurement or process noise on the estimation of the input distribution, and let hope for future extensions in a full non-linear framework, for example through the use of linear Gaussian mixtures.

翻译：随机反演问题通常在需要量化计算机模型输入所影响的不确定性时遇到。这类问题涉及根据含噪声的可观测输出估计输入分布，并在贝叶斯背景下得到越来越多的研究，其中目标输入受到随机不确定性的影响。在此意义上，可称一个随机输入是有意义的，前提是它能解释大部分输出不确定性。尽管这类逆问题具有可辨识性条件，但在模型定义阶段、推断之前，应考虑能够形式化这种意义的"信号噪声比"约束。本文探讨了在参数不确定性量化背景下，通过全局敏感性分析与信息论工具（方差、熵、费舍尔信息）强制解具有意义的可能性。此类强制主要体现为对输入协方差施加约束，并可通过考虑线性或可线性化模型使其显式化。模拟实验表明，当这些约束被注入建模过程时，能够限制测量噪声或过程噪声对输入分布估计的影响，并为未来在完全非线性框架下的扩展（例如通过使用线性高斯混合模型）提供了希望。