Comparison of methodologies to assess the convergence of Markov chain Monte Carlo methods

One major challenge with the modelization of complex problems using Markov chain Monte Carlo (MCMC) methods is the determination of the length of the chain in order to reach convergence. This paper is devoted to parametric empirical methods testing the stationarity. We compare the methods of Gelman and Rubin, Yu and Mykland, Raftery and Lewis, Geweke, Riemann sums and the subsampling. These methods are tested using three examples: the simple case of the generation of a normal random variable, a bivariate mixture of normal models and a practical case taken from hydrology, namely the shifting level model. Results show that no method works in every case. We therefore suggest a joint use of these techniques. The importance of determining carefully the burn-in period is also highlighted.