## making a random walk geometrically ergodic

**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.

March 3, 2013 at 3:01 pm

Hmmm. I wonder if this also says stuff about MALA. Given that the analysis Yves Atchade did on the geometric ergodicity of MALA in that adaptive paper was built off the proof method of Jarner and Hansen, everything here probably applies. And I would strongly suspect that this would give a way to look at the geometric ergodicity of manifold MALA without having to spend your life staring at drift functions….

(given that the ”manifold” in question is quite boring as there is a global chart, you could almost certainly make some sort of link to changes of variables. See, for instance, the two ways of turning an isotropic random field into a non-isotropic one: you can either define it on an explicitly deformed space and map it back to R^2 [Samson and Guttorp] or you can intrinsically define a metric through a spatially varying 2×2 positive definite matrix [Lindgren, Rue, Lindström])