Archive for uniform ergodicity

approximations of Markov Chains [another garden of forking paths]

Posted in Books, Mountains, pictures, Statistics, University life with tags , , , , , , , , , , on March 15, 2016 by xi'an

On the Sétaz cabin ride, Valloire, Dec. 23, 2011James Johndrow and co-authors from Duke wrote a paper on approximate MCMC that was arXived last August and that I missed. David Dunson‘s talk at MCMski made me aware of it. The paper studies the impact of replacing a valid kernel with a close approximation. Which is a central issue for many usages of MCMC in complex models, as exemplified by the large number of talks on that topic at MCMski.

“All of our bounds improve with the MCMC sample path length at the expected rate in t.”

A major constraint in the paper is Doeblin’s condition, which implies uniform geometric ergodicity. Not only it is a constraint on the Markov kernel but it is also one for the Markov operator in that it may prove impossible to… prove. The second constraint is that the approximate Markov kernel is close enough to the original, which sounds reasonable. Even though one can always worry that the total variation norm is too weak a norm to mean much. For instance, I presume with some confidence that this does not prevent the approximate Markov kernel from not being ergodic, e.g., not irreducible, not absolutely continuous wrt the target, null recurrent or transient. Actually, the assumption is stronger in that there exists a collection of approximations for all small enough values ε of the total variation distance. (Small enough meaning ε is much smaller than the complement α to 1 of the one step distance between the Markov kernel and the target. With poor kernels, the approximation must thus be very good.) This is less realistic than assuming the availability of one single approximation associated with an existing but undetermined distance ε. (For instance, the three examples of Section 3 in the paper show the existence of approximations achieving a certain distance ε, without providing a constructive determination of such approximations.) Under those assumptions, the average of the sequence of Markov moves according to the approximate kernel converges to the target in total variation (and in expectation for bounded functions). With sharp bounds on those distances. I am still a bit worried at the absence of conditions for the approximation to be ergodic.

“…for relatively short path lengths, there should exist a range of values for which aMCMC offers better performance in the compminimax sense.”

The paper also includes computational cost into the picture. Introducing the notion of compminimax error, which is the smallest (total variation) distance among all approximations at a given computational budget. Quite an interesting, innovative, and relevant notion that may however end up being too formal for practical use. And that does not include the time required to construct and calibrate the approximations.

Advances in scalable Bayesian computation [day #1]

Posted in Books, Mountains, pictures, R, Statistics, University life with tags , , , , , , , , , on March 4, 2014 by xi'an

polyptych painting within the TransCanada Pipeline Pavilion, Banff Centre, Banff, March 21, 2012This was the first day of our workshop Advances in Scalable Bayesian Computation and it sounded like the “main” theme was probabilistic programming, in tune with my book review posted this morning. Indeed, both Vikash Mansinghka and Frank Wood gave talks about this concept, Vikash detailing the specifics of a new programming language called Venture and Frank focussing on his state-space version of the above called Anglican. This is a version of the language Church, developed to handle probabilistic models and inference (hence the joke about Anglican, “a Church of England Venture’! But they could have also added that Frank Wood was also the name of a former archbishop of Melbourne..!) I alas had an involuntary doze during Vikash’s talk, which made it harder for me to assess the fundamentals of those ventures, of how they extended beyond a “mere” new software (and of why I would invest in learning a Lisp-based language!).

The other talks of Day #1 were of a more “classical” nature with Pierre Jacob explaining why non-negative unbiased estimators were impossible to provide in general, a paper I posted about a little while ago, and including an objective Bayes example that I found quite interesting. Then Sumeet Singh (no video) presented a joint work with Nicolas Chopin on the uniform ergodicity of the particle Gibbs sampler, a paper that I should have commented here (except that it appeared just prior to The Accident!), with a nice coupling proof. And Maria Lomeli gave us an introduction to the highly general Poisson-Kingman mixture models as random measures, which encompasses all of the previously studied non-parametric random measures, with an MCMC implementation that included a latent variable representation for the alpha-stable process behind the scene, representation that could be (and maybe is) also useful in parametric analyses of alpha-stable processes.

We also had an open discussion in the afternoon that ended up being quite exciting, with a few of us voicing out some problems or questions about existing methods and others making suggestions or contradictions. We are still a wee bit short of considering a collective paper on MCMC under constraints with coherent cross-validated variational Bayes and loss-based pseudo priors, with applications to basketball data” to appear by the end of the week!

Add to this two visits to the Sally Borden Recreation Centre for morning swimming and evening climbing, and it is no wonder I woke up a bit late this morning! Looking forward Day #2!

making a random walk geometrically ergodic

Posted in R, Statistics with tags , , , , , , , , , on March 2, 2013 by xi'an

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.

How quickly does randomness appear?

Posted in Statistics, University life with tags , , , , , , , , on November 10, 2011 by xi'an

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…

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