Pitfalls of Markov chain Monte-Carlo techniques

One can doubt that maximum-likelihood algorithms always find the true global maximum of the likelihood function. Similarly, with MCMC techniques, the Markov chain can fail to converge to the stationary distribution of the posterior probabilities. A possible reason for this is the failure to visit all highly probable regions of the parameter space because of local maxima in the likelihood curve. However poor proposal mechanisms and/or failure to run the chain long enough are usually the main cause of sample defect  (see Huelsenbeck et al., 2002). Unfortunately it is not always easy to identify these traps. We can only recommend to do long runs, monitor the convergence of several model parameters since monitoring the likelihood only is not enough, and repeat the experiment using different random starting trees to check that all the chains give similar results (i.e., substitution model parameters, consensus tree, likelihood, ...).