Assessment of MCMC convergence: a time series and dynamical systems approach

Important in the application of Markov chain Monte Carlo (MCMC) methods is the determination that a search run has converged. Given that such searches typically take place in high-dimensional spaces, there are many pitfalls and difficulties in making such assessments. We discuss the use of phase randomisation as tool in the MCMC context, provide some details of its distributional properties for time series which enable its use as a convergence diagnostic, and contrast its performance with a selection of other widely used diagnostics. Some comments on analytical results, obtained via Edgeworth expansion, are also made.