  Over the last decade, a series of applied mathematics papers have explored a
type of inverse problem--called by a variety of names including "inverse
sensitivity", "pushforward based inference", "consistent Bayesian inference",
or "data-consistent inversion"--wherein a solution is a probability density
whose pushforward takes a given form. The formulation of such a stochastic
inverse problem can be unexpected or confusing to those familiar with
traditional Bayesian or otherwise statistical inference. To date, two classes
of solutions have been proposed, and these have only been justified through
applications of measure theory and its disintegration theorem. In this work we
show that, under mild assumptions, the formulation of and solution to all
stochastic inverse problems can be more clearly understood using basic
probability theory: a stochastic inverse problem is simply a
change-of-variables or approximation thereof. For the two existing classes of
solutions, we derive the relationship to change(s)-of-variables and illustrate
using analytic examples where none had previously existed. Our derivations use
neither Bayes' theorem nor the disintegration theorem explicitly. Our final
contribution is a careful comparison of changes-of-variables to more
traditional statistical inference. While taking stochastic inverse problems at
face value for the majority of the paper, our final comparative discussion
gives a critique of the framework.
