Bayesians have their own ways

IDRS - 2
Neelabh Rohan on Information Criteria for Bayesian's

Under classical setup for choosing models we use AIC and BIC criteria. But these involve likelihood which further takes the parameter as constant. But if we are defining model with prior distribution for parameter itself (that is where Bayesian comes into picture), then we cannot use the usual likelihood and hence criteria fail.

The first one is Deviance Information Criteria, given by Spiegelhalter (2000) in JRSS, which considers the model complexity (in terms of number of parameters) at the same time the usual -2log L term (which involves posterior estimate of the parameter).

The Focused Information criteria, given by Claskens and Hjort (2003) in JASA, considers only those parameters (or their functions) which are in focus, instead of taking all. This criteria is there for both frequentists as well as Bayesians. We have some fixed parameters (narrow part of model) and some parameters which keep changing (the wide part of it). So using the information matrices for these vector parameters a criteria is being formulated.

PS 1: The beauty of these criterias is that they can handle the models which have varying number of parameters.
PS 2: I will miss the next session. :(