  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.
