  In this article, we propose a novel Bayesian multiple testing formulation for
model and variable selection in inverse setups, judiciously embedding the idea
of inverse reference distributions proposed by Bhattacharya (2013) in a mixture
framework consisting of the competing models. We develop the theory and methods
in the general context encompassing parametric and nonparametric competing
models, dependent data, as well as misspecifications. Our investigation shows
that asymptotically the multiple testing procedure almost surely selects the
best possible inverse model that minimizes the minimum Kullback-Leibler
divergence from the true model. We also show that the error rates, namely,
versions of the false discovery rate and the false non-discovery rate converge
to zero almost surely as the sample size goes to infinity. Asymptotic
{\alpha}-control of versions of the false discovery rate and its impact on the
convergence of false non-discovery rate versions, are also investigated.
  Our simulation experiments involve small sample based selection among inverse
Poisson log regression and inverse geometric logit and probit regression, where
the regressions are either linear or based on Gaussian processes. Additionally,
variable selection is also considered. Our multiple testing results turn out to
be very encouraging in the sense of selecting the best models in all the
non-misspecified and misspecified cases.
