  A recently developed measure-theoretic framework solves a stochastic inverse
problem (SIP) for models where uncertainties in model output data are
predominantly due to aleatoric (i.e., irreducible) uncertainties in model
inputs (i.e., parameters). The subsequent inferential target is a distribution
on parameters. Another type of inverse problem is to quantify uncertainties in
estimates of "true" parameter values under the assumption that such
uncertainties should be reduced as more data are incorporated into the problem,
i.e., the uncertainty is considered epistemic. A major contribution of this
work is the formulation and solution of such a parameter identification problem
(PIP) within the measure-theoretic framework developed for the SIP. The
approach is novel in that it utilizes a solution to a stochastic forward
problem (SFP) to update an initial density only in the parameter directions
informed by the model output data. In other words, this method performs
"selective regularization" only in the parameter directions not informed by
data. The solution is defined by a maximal updated density (MUD) point where
the updated density defines the measure-theoretic solution to the PIP. Another
significant contribution of this work is the full theory of existence and
uniqueness of MUD points for linear maps with Gaussian distributions.
Data-constructed Quantity of Interest (QoI) maps are also presented and
analyzed for solving the PIP within this measure-theoretic framework as a means
of reducing uncertainties in the MUD estimate. We conclude with a demonstration
of the general applicability of the method on two problems involving either
spatial or temporal data for estimating uncertain model parameters.
