Comparing partition and mixture models with Akaike information criteria

Sophisticated phylogenetic models often include mixture and/or partition
model components. It was recently noted that information criteria tend to
favour partition models over mixture models even in some cases where the
latter are misspecified and give poor topological estimation. We show that
this problem arises because partition models and mixture models
fundamentally differ in their probability calculations: mixture models
calculate site-wise likelihoods as the marginal probability of the data
averaging over parameter vectors that might have arisen at a site whereas
partition model site likelihoods are calculated as the probability of the
site pattern conditional upon a fixed assigned parameter vector at that
site. These differing probability calculations lead to AIC estimates that
are not comparable. We explore three generally applicable ways of
correcting the issue.