Specification of additional information for solving stochastic inverse problems

Methods have been developed to identify the probability distribution of a random vector $Z$ from information consisting of its bounded range and the probability density function or moments of a quantity of interest, $Q(Z)$. The mapping from $Z$ to $Q(Z)$ may arise from a stochastic differential equation whose coefficients depend on $Z$. This problem differs from Bayesian inverse problems as the latter is primarily driven by observation noise. We motivate this work by demonstrating that additional information on $Z$ is required to recover its true law. Our objective is to identify what additional information on $Z$ is needed and propose methods to recover the law of $Z$ under such information. These methods employ tools such as Bayes' theorem, principle of maximum entropy, and forward uncertainty quantification to obtain solutions to the inverse problem that are consistent with information on $Z$ and $Q(Z)$. The additional information on $Z$ may include its moments or its family of distributions. We justify our objective by considering the capabilities of solutions to this inverse problem to predict the probability law of unobserved quantities of interest.