  In this article, we investigate posterior convergence in nonparametric
regression models where the unknown regression function is modeled by some
appropriate stochastic process. In this regard, we consider two setups. The
first setup is based on Gaussian processes, where the covariates are either
random or non-random and the noise may be either normally or
double-exponentially distributed. In the second setup, we assume that the
underlying regression function is modeled by some reasonably smooth, but
unspecified stochastic process satisfying reasonable conditions. The
distribution of the noise is also left unspecified, but assumed to be
thick-tailed. As in the previous studies regarding the same problems, we do not
assume that the truth lies in the postulated parameter space, thus explicitly
allowing the possibilities of misspecification. We exploit the general results
of Shalizi (2009) for our purpose and establish not only posterior consistency,
but also the rates at which the posterior probabilities converge, which turns
out to be the Kullback-Leibler divergence rate. We also investigate the more
familiar posterior convergence rates. Interestingly, we show that the posterior
predictive distribution can accurately approximate the best possible predictive
distribution in the sense that the Hellinger distance, as well as the total
variation distance between the two distributions can tend to zero, in spite of
misspecifications.
