The Gelman-Rubin convergence diagnostic for Markov chain Monte Carlo methods

The goal of this thesis is showing practical examples of comparison between the diagnostic of the convergence of Markov chain Monte Carlo algorithms, conceived by Gelman and Rubin (1992), and its upgrade elaborated by Vats and Knudson (2021). In order to achieve this purpose it was necessary firstly to recall some notions about the Markov chains - the discussion is made in general space setting; secondly to describe some MCMC algorithms, again with theoretical justifications. The first chapter covers the part about Markov chains according to the book by Robert and Casella (2004). The transition kernels and Markov chains are defined in general spaces, then the concepts of irreducibility and aperiodicity are handled, in order to arrive to recurrence and Harris recur- rence. Then there are the most important sections introducing the concept of ergodicity and enunciating the limit theorems that are used in the fol- lowing chapter. The second chapter introduces the Monte Carlo method, arriving to for- malize the concept of Markov chain Monte Carlo. The discussion, again, follows Robert and Casella (2004). Thus, after the explanation about the reasons why MCMC methods are so important, the chapter continues the discussion with the most important MCMC: Metropolis Hastings and Gibbs sampler. The third - and most interesting - chapter recovers extracts by Gelman and Rubin (1992) and Vats and Knudson (2021) in order to present the two statistics used as diagnostics of the convergence and their differences. The last section, with my own simulations, aims to clarify with graphics and comments the meaning of what is theoretically exposed by the authors of the papers.