**W**hile a random walk Metropolis-Hastings algorithm cannot be uniformly ergodic in a general setting (Mengersen and Tweedie, *AoS*, 1996), because it needs more energy to leave far away starting points, it can be geometrically ergodic depending on the target (and the proposal). In a recent *Annals of Statistics* paper, Leif Johnson and Charlie Geyer designed a trick to turn a random walk Metropolis-Hastings algorithm into a geometrically ergodic random walk Metropolis-Hastings algorithm by virtue of an isotropic transform (under the provision that the original target density has a moment generating function). This theoretical result is complemented by an R package called mcmc. (I have not tested it so far, having read the paper in the métro.) The examples included in the paper are however fairly academic and I wonder how the method performs in practice, on truly complex models, in particular because the change of variables relies on (a) an origin and (b) changing the curvature of space uniformly in all dimensions. Nonetheless, the idea is attractive and reminds me of a project of ours with Randal Douc, started thanks to the ‘Og and still under completion.

## Archive for geometric ergodicity

## making a random walk geometrically ergodic

Posted in R, Statistics with tags Annals of Statistics, CRAN, geometric ergodicity, métro, MCMC, Metropolis-Hastings, R, R package, random walk, uniform ergodicity on March 2, 2013 by xi'an## dirty MCMC streams

Posted in Statistics, Travel, University life with tags geometric ergodicity, Markov chain Monte Carlo, MCMC, pseudo-random generator, randomness on May 7, 2012 by xi'an

**I**ain Murray and Lloyd T. Elliott had posted this paper on arXiv just before I left for my U,K, 2012 tour and I did not have time to read it in detail, nor obviously to report on it. Fortunately, during the ICMS meeting, Iain presented an handmade poster on this paper that allowed me a quick tour, enough to report on the contents! The main point of the paper is that it is possible to modify many standard MCMC codes so that they can be driven by a dependent random sequence. The authors show that various if specific dependent sequences of uniform variates do not modify the right target and the ergodicity of the MCMC scheme. As mentioned in the conclusion of the paper, this may have interesting consequences in parallel implementations where randomness becomes questionable, or in physical random generators, whose independence may also be questionable…

## How quickly does randomness appear?

Posted in Statistics, University life with tags convergence, geometric ergodicity, La Recherche, Metropolis-Hastings algorithms, Nice, randomness, spectral gap, total variation, uniform ergodicity on November 10, 2011 by xi'an**T**his was the [slightly off-key] title of the math column in the November issue of *La Recherche*, in any case intriguing enough for me to buy this general public science magazine on the metro platform and to read it immediately while waiting for an uncertain train, thanks to the nth strike of the year on my train line… But this was the occasion for an exposition of the Metropolis algorithm in a general public journal! The column actually originated from a recently published paper by Persi Diaconis, Gilles Lebeaux, and Laurent Michel, *Geometric analysis for the Metropolis algorithm on Lipschitz domain*, in ** Inventiones Mathematicae** [one of the top pure math journals]. The column in

*La Recherche*described the Metropolis algorithm (labelled there a

*random walk on Markov chains*!), alluded to the use of MCMC methods in statistics, told the genesis of the paper [namely the long-term invitation of Persi Diaconis in Nice a few years ago] and briefly explained the convergence result, namely the convergence of the Metropolis algorithm to the stationary measure at a geometric rate, with an application to the non-overlapping balls problem.

**I**f you take a look at the paper, you will see it is a beautiful piece of mathematics, establishing a spectral gap on the Markov operator associated with the Metropolis algorithm and deducing a uniformly geometric convergence [in total variation] for most regular-and-bounded-support distributions. A far from trivial and fairly general result. *La Recherche* however fails to mention the whole corpus of MCMC convergence results obtained in the 1990’s and 2000’s, by many authors, incl. Richard Tweedie, Gareth Roberts, Jeff Rosenthal, Eric Moulines, Gersende Fort, Randal Douc, Kerrie Mengersen, and others…

## Random dive MH

Posted in R, Statistics with tags geometric ergodicity, Metropolis-Hastings, R, random walk, randomness on September 2, 2010 by xi'an**A** new Metropolis-Hastings algorithm that I would call “universal” was posted by Somak Dutta yesterday on arXiv. *Multiplicative random walk Metropolis-Hastings on the real line* contains a different Metropolis-Hastings algorithm called the *random dive*. The proposed new value *x’* given the current value *x* is defined by

when is a random variable on . Thus, at each iteration, the current value is either shrunk or expanded by a random amount. When I read the paper in the Paris metro, while appreciating that this algorithm could be geometrically ergodic as opposed to the classical random walk algorithm, I was not convinced by the practical impact of the method, as I thought that the special role of zero in the proposal was not always reflected in the target. Especially when considering that the proposal is parameter-free. However, after running the following R program on a target centred away from zero, I found the proposal quite efficient.

f=function(x){.5*dnorm(x,mean=14)+.5*dnorm(x,mean=35)}

Nsim=10^5

x=rep(5,Nsim)

for (t in 2:Nsim){

coef=runif(1,min=-1)^sample(c(-1,1),1)

prop=x[t-1]*coef

prob=abs(coef)*f(prop)/f(x[t-1])

x[t]=x[t-1]

if (runif(1)<prob) x[t]=prop

}

hist(x,pro=T,nclass=113,col=”wheat2″)

curve(f,add=T,n=1001,col=”sienna”,lwd=2)

Obviously, it is difficult to believe that this extension will keep working similarly well when the dimension increases but this is an interesting way of creating a heavy tail proposal.