  Comparison of competing statistical models is an essential part of
psychological research. From a Bayesian perspective, various approaches to
model comparison and selection have been proposed in the literature. However,
the applicability of these approaches strongly depends on the assumptions about
the model space $\mathcal{M}$, the so-called model view. Furthermore,
traditional methods like leave-one-out cross-validation (LOO-CV) estimate the
expected log predictive density (ELPD) of a model to investigate how the model
generalises out-of-sample, which quickly becomes computationally inefficient
when sample size becomes large. Here, we provide a tutorial on approximate
Pareto-smoothed importance sampling leave-one-out cross-validation (PSIS-LOO),
a computationally efficient method for Bayesian model comparison. First, we
discuss several model views and the available Bayesian model comparison methods
in each. We then use Bayesian logistic regression as a running example how to
apply the method in practice, and show that it outperforms other methods like
LOO-CV or information criteria in terms of computational effort while providing
similarly accurate ELPD estimates. In a second step, we show how even
large-scale models can be compared efficiently by using posterior
approximations in combination with probability-proportional-to-size
subsampling. We show how to compare competing models based on the ELPD
estimates provided, and how to conduct posterior predictive checks to safeguard
against overconfidence in one of the models under consideration. We conclude
that the method is attractive for mathematical psychologists who aim at
comparing several competing statistical models, which are possibly
high-dimensional and in the big-data regime.
