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 likelihood contributions from sites as the marginal probability of the data averaging over parameter vectors that might have arisen at a site whereas partition model likelihood contributions 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 always comparable. We explore three generally applicable ways of correcting the issue.