  We consider Bayesian inference in inverse regression problems where the
objective is to infer about unobserved covariates from observed responses and
covariates. We establish posterior consistency of such unobserved covariates in
Bayesian inverse regression problemsunder appropriate priors in a leave-one-out
cross-validation setup. We relate this to posterior consistency of inverse
reference distributions (Bhattacharya (2013)) for assessing model adequacy. We
illustrate our theory and methods with various examples of Bayesian inverse
regression, along with adequate simulation experiments.
