  We review the Akaike, deviance, and Watanabe-Akaike information criteria from
a Bayesian perspective, where the goal is to estimate expected
out-of-sample-prediction error using a biascorrected adjustment of
within-sample error. We focus on the choices involved in setting up these
measures, and we compare them in three simple examples, one theoretical and two
applied. The contribution of this review is to put all these information
criteria into a Bayesian predictive context and to better understand, through
small examples, how these methods can apply in practice.
