Complex models of sequence evolution require accurate estimators as exemplified with the invariable site plus Gamma model

The invariable site plus Γ model is widely used to model rate heterogeneity among alignment sites in maximum likelihood and Bayesian phylogenetic analyses. The proof that the invariable site plus continuous Γ model is identifiable (model parameters can be inferred correctly given enough data) has increased the creditability of its application to phylogeny reconstruction. However, most phylogenetic software implement the invariable site plus discrete Γ model, whose identifiability is likely but unproven. How well the parameters of the invariable site plus discrete Γ model are estimated is still disputed. Especially the correlation of the fraction of invariable sites with the fractions of sites with a slow evolutionary rate is discussed as being problematic. We show that optimization heuristics as implemented in frequently used phylogenetic software cannot always reliably estimate the shape parameter, the proportion of invariable sites and the tree length. Here, we propose an improved opt...