  We formulate, and present a numerical method for solving, an inverse problem
for inferring parameters of a deterministic model from stochastic observational
data (quantities of interest). The solution, given as a probability measure, is
derived using a Bayesian updating approach for measurable maps that finds a
posterior probability measure, that when propagated through the deterministic
model produces a push-forward measure that exactly matches the observed
probability measure on the data. Our approach for finding such posterior
measures, which we call consistent Bayesian inference, is simple and only
requires the computation of the push-forward probability measure induced by the
combination of a prior probability measure and the deterministic model. We
establish existence and uniqueness of observation-consistent posteriors and
present stability and error analysis. We also discuss the relationships between
consistent Bayesian inference, classical/statistical Bayesian inference, and a
recently developed measure-theoretic approach for inference. Finally,
analytical and numerical results are presented to highlight certain properties
of the consistent Bayesian approach and the differences between this approach
and the two aforementioned alternatives for inference.
