While 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 uniform ergodicity
This 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.
If 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…