  A natural method for approximating out-of-sample predictive evaluation is
leave-one-out cross-validation (LOOCV) --- we alternately hold out each case
from a full data set and then train a Bayesian model using Markov chain Monte
Carlo (MCMC) without the held-out; at last we evaluate the posterior predictive
distribution of all cases with their actual observations. However, actual LOOCV
is time-consuming. This paper introduces two methods, namely iIS and iWAIC, for
approximating LOOCV with only Markov chain samples simulated from a posterior
based on a \textit{full} data set. iIS and iWAIC aim at improving the
approximations given by importance sampling (IS) and WAIC in Bayesian models
with possibly correlated latent variables. In iIS and iWAIC, we first integrate
the predictive density over the distribution of the latent variables associated
with the held-out without reference to its observation, then apply IS and WAIC
approximations to the integrated predictive density. We compare iIS and iWAIC
with other approximation methods in three real data examples that respectively
use mixture models, models with correlated spatial effects, and a random effect
logistic model. Our empirical results show that iIS and iWAIC give
substantially better approximates than non-integrated IS and WAIC and other
methods.
