On model selection and concavity for finite mixture models

We show that the log-likelihood of finite mixture models is approximately concave as a function of the number of mixture components k. A corollary of this result is that the penalized log-likelihood will also be approximately concave (as a function of k) if the penalty term is itself strictly concave or linear in k (true, for example, for BIC). These results have a number of significant practical implications for parameter estimation and model selection in a mixture context