**A Simple Convergence Diagnostic for Markov Chain
Monte Carlo**

E. Gutiérrez-Peña and J.M. González-Barrios

*IIMAS-UNAM, Mexico*

**Abstract. **Markov Chain Monte
Carlo techniques draw samples from a multivariate density *p*(
*t*_1 , . . . , *t*_*d *) by simulating a Markov chain
whose equilibrium distribution is precisely *p*(
*t*_1 , . . . , *t*_*d *). However, although
convergence is generally well understood theoretically, diagnosis of
convergence is often a difficult task. In practice there are two
sides to convergence. First we would like to find all areas of
non-negligible probability. Secondly, we would like to obtain enough
samples to accurately estimate quantities of interest. It has been
argued that the assessment of convergence should be relative to what
we wish to achieve by the analysis. Hence most of the convergence
diagnostics proposed in the literature focus on the distribution of
some real-valued function *r* = *g*( *t*_1
, . . . , *t*_*d *). It must be pointed out,
however, that in general one is trying to assess the convergence of
all *d *random variables to the joint distribution and so care
must be taken when only one dimensional summaries are examined. In
this paper we propose a new convergence diagnostic based on a
multivariate nonparametric two-sample test.

*Key words:* Bayesian inference; convergence of stochastic
processes; Gibbs sampler; Metropolis-Hastings algorithm.