Heterotachy, the variation of substitution rate at a site across time, is a 
prevalent phenomenon in nucleotide and amino acid alignments, which may mislead 
probabilistic-based phylogenetic inferences. The covarion model is a special 
case of heterotachy, in which sites change between the "ON" state (allowing 
substitutions according to any particular model of sequence evolution) and the 
"OFF" state (prohibiting substitutions). In current implementations, the switch 
rates between ON and OFF states are homogeneous across sites, a hypothesis that 
has never been tested. In this study, we developed an infinite mixture model, 
called the covarion mixture (CM) model, which allows the covarion parameters to 
vary across sites, controlled by a Dirichlet process prior. Moreover, we combine 
the CM model with other approaches. We use a second independent Dirichlet 
process that models the heterogeneities of amino acid equilibrium frequencies 
across sites, known as the CAT model, and general rate-across-site heterogeneity 
is modeled by a gamma distribution. The application of the CM model to several 
large alignments demonstrates that the covarion parameters are significantly 
heterogeneous across sites. We describe posterior predictive discrepancy tests 
and use these to demonstrate the importance of these different elements of the 
models.