  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.
