Inequalities between expected marginal log-likelihoods, with implications for likelihood-based model complexity and comparison measures

A multi-level model allows the possibility of marginalization across levels in different ways, yielding more than one possible marginal likelihood. Since log-likelihoods are often used in classical model comparison, the question to ask is which likelihood should be chosen for a given model. The authors employ a Bayesian framework to shed some light on qualitative comparison of the likelihoods associated with a given model. They connect these results to related issues of the effective number of parameters, penalty function, and consistent definition of a likelihood-based model choice criterion. In particular, with a two-stage model they show that, very generally, regardless of hyperprior specification or how much data is collected or what the realized values are, a priori, the first-stage likelihood is expected to be smaller than the marginal likelihood. A posteriori, these expectations are reversed and the disparities worsen with increasing sample size and with increasing number of model levels.