  In the Bayesian literature on model comparison, Bayes factors play the
leading role. In the classical statistical literature, model selection criteria
are often devised used cross-validation ideas. Amalgamating the ideas of Bayes
factor and cross-validation Geisser and Eddy (1979) created the pseudo-Bayes
factor. The usage of cross-validation inculcates several theoretical
advantages, computational simplicity and numerical stability in Bayes factors
as the marginal density of the entire dataset is replaced with products of
cross-validation densities of individual data points.
  However, the popularity of pseudo-Bayes factors is still negligible in
comparison with Bayes factors, with respect to both theoretical investigations
and practical applications. In this article, we establish almost sure
exponential convergence of pseudo-Bayes factors for large samples under a
general setup consisting of dependent data and model misspecifications. We
particularly focus on general parametric and nonparametric regression setups in
both forward and inverse contexts.
  We illustrate our theoretical results with various examples, providing
explicit calculations. We also supplement our asymptotic theory with simulation
experiments in small sample situations of Poisson log regression and geometric
logit and probit regression, additionally addressing the variable selection
problem. We consider both linear and nonparametric regression modeled by
Gaussian processes for our purposes. Our simulation results provide quite
interesting insights into the usage of pseudo-Bayes factors in forward and
inverse setups.
