Stochastic inverse 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：随机逆问题通常出现在需要量化影响计算机模型输入的不确定性时。这类问题涉及从带噪声的可观测输出中估计输入分布，并且在贝叶斯框架下日益受到关注，其中目标输入受随机不确定性影响。在此背景下，若某个随机输入能解释大部分输出不确定性，则可被定义为有意义的。虽然此类逆问题具有可辨识性条件，但必须在模型定义阶段、在推断之前，将能够形式化这种意义性的“信噪比”约束纳入考虑。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和信息论（方差、熵、费舍尔信息）工具，迫使解具有意义性的可能性。此类强制措施主要体现为对输入协方差的约束，并且可通过考虑线性或可线性化模型来明确表达。模拟实验表明，当这些约束被引入建模过程时，能够限制测量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中的扩展（例如通过线性高斯混合模型）提供了可能。