sampling using adaptive regenerative processes

We just posted a new arXival on Sampling using Adaptive Regenerative Processes, written by Hector McKimm (Warwick), Andi Wang (soon Warwick), Murray Pollock (ex-Warwick), Gareth Roberts (Warwick) and myself. This is a collaborative that has been going on for a while, mostly via zoom in these Covid times. It builds upon the earlier paper of Wang et al.  (2021) constructing the regeneration process (Restore), by aiming at improving this process by adapting the regeneration distribution and hence dramatically reducing the number of regenerations. Gaining in addition the ability to sample from target distributions for which simulation under a fixed regeneration distribution is computationally intractable. This work is part of Hector’s PhD, written at Warwick.

Kick-Kac teleportation

Randal Douc, Alain Durmus, Aurélien Enfroy, and Jimmy Olson have arXived their Kick-Kac teleportation paper, which was presented by Randal at CIRM last semester. It is based on Kac’s theorem, which states that, for a Markov chain with invariant distribution π, under (π) stationarity, the average tour between two visits to an accessible set C is also stationary. Which can be used for approximating π(h) if π(C) is known (or well-estimated). Jim Hobert and I exploited this theorem in our 2004 perfect sampling paper. The current paper contains a novel proof of the theorem under weaker conditions. (Note that the only condition on C is that it is accessible, rather than a small set. Which becomes necessary for geometric ergodicity, see condition (A4).)

What they define as the Kick-Kac teleportation (KKT) process is the collection of trajectories between two visits to C. Their memoryless version requires perfect simulations from π restricted to the set C. With a natural extension based on a Markov kernel keeping π restricted to the set C stationary. And a further generalisation allowing for lighter tails that also contains the 2005 paper by Brockwell and Kadane as a special case.

The ability of generating from a different kernel Q at each visit to C allows for different dynamics (as in other composite kernels). In their illustrations, the authors use lowest density regions for C, which is rather surprising to me. Except that it allows for a better connection between modes of the target π: the higher performances of the KKT algorithms against the considered alternatives are apparently dependent on the ability of the kernel Q to explore other modes with sufficient frequency.

Statistics, with interactions

Due to a tight June schedule (3rd conference in a week!), I only stayed one day at the SIOD 2013 conference in Saigon. (SIOD means Statistics and interaction with other disciplines.) The conference was housed by Ton Duc Thang University, on a very modern campus, and it sounded like the university had drafted a lot of his undergrads to catter to the SIOD participants: similar to the Bayesian conference in India a few months ago, those students would stand at the ready to guide us around the campus and to relay any problem to the organisers. This was very helpful and enjoyable, a plus being that most female students wore the traditional pink costume adopted by the university, but it also made me a wee bit uncomfortable as I do not know how much say those students had in this draft… In particular, most of the students I talked with were from other fields than Statistics. (And definitely not complaining, but being on the opposite very friendly the whole time!) A funny side story is that I got a wake-up call from the conference organisers in the morning as I had missed a welcome ceremony with the president due to oversleeping (itself due to an excess of iced coffee rather than minimal jetlag!). Among the few talks I attended, some French school statistics due to the presence of a large contingent from Toulouse, a talk about zero inflated normal distributions which sounded like missing-at-random normal observations (hence easy to process), and a talk about the point of using Bayes factors in hypothesis testing which essentially if independently provided a second version of my course from the previous day.

Yesterday, I also had a short discussion with Paul Minh who presented a talk on a general regenerative device for MCMC algorithms, using a bound on the target density rather than on the Markov transition in order to achieve easier regeneration. While a neat idea, this method requires the construction of a lower bound that can easily simulated. Furthermore, if the regeneration probability is low, the mixing speed may remain similar to the original MCMC sampler, as the method ressorts to a standard MCMC step on the remaining part of the target density.

advanced Markov chain Monte Carlo methods

This book, Advanced Markov Chain Monte Carlo Methods: Learning from Past Samples, by Faming Liang, Chuanhai Liu, and Raymond Carroll, appeared last year and has been sitting on my desk all this time, patiently (?) waiting for a review. When I received it, I took a brief look at it (further than the cool cover!) and then decided I needed more than that to write a useful review! Here are my impressions  on Advanced Markov Chain Monte Carlo Methods after a deeper read. (I have not read any other review in the main statistical journals so far.)

The title, Advanced Markov Chain Monte Carlo Methods, is a clear warning on the level of the book: “advanced”, it certainly is!!! By page 85, the general description of MCMC simulation methods is completed, including perfect sampling and reversible jump MCMC, and the authors engage into a detailed description of highly specialised topics of their choice: Auxiliary variables (Chap. 4), Population-based MCMC (Chap. 5), Dynamic weighting (Chap. 6), Stochastic approximation Monte Carlo (Chap. 7), and MCMC with adaptive proposals (Chap. 8).  The book is clearly inspired by the numerous papers the authors have written in those area, especially Faming Liang. (The uneven distribution of the number of citations per year with peaks in 2000 and 2009 reflects this strong connection.) While the book attempts at broadening the spectrum by including introductory sections, and discussing other papers, it remains nonetheless that this centred focus of the book reduces its potential readership to graduate students and researchers who could directly work on the original papers. I would thus hesitate in teaching my graduate students from this book, given that they only attend a single course on Monte Carlo methods.

Adap’skiii [day 2]

Another exciting day at Adap’skiii!!!

Yves Atchadé presented a very recent work on the fundamental issue of estimating the asymptotic variance estimation for adaptive MCMC algorithms, with an intriguing experimental observation that a non-converging bandwidth with rate 1/n was providing better coverage than the converging rate. (I always found the issue of estimating the asymptotic variance both a tough problem and an important item in convergence assessment.) Galin Jones showed new regeneration results for componentwise MCMC samplers, with applications to quantile estimation. The iid structure produced by the regeneration mechanism allows rather naturally to introduce an adaptive improvement in those algorithms, if regeneration occurs often enough. (From the days of my Stat’Sci’ paper on convergence assessment, I  love regeneration techniques for both theoretical and methodological reasons, even though they are often difficult to efficiently implement in practice.) Matti Vihola summarised several of his recent papers on the stability and convergence of adaptive MCMC algorithms, pursuing the Finnish tradition of leadership in adaptive algorithms! One point I found particularly interesting was the possibility of separating ergodicity from the Law of Large Numbers, thus reducing the constraints imposed by the containment condition. In the afternoon, Dawn Woodard discussed the convergence rate of the Gibbs sampler used for genomic motif discovery by Liu, Lawrence and Neuwald (1995). Scott Schmidler concluded the workshop by a far-ranging talk distinguishing between exploration and exploitation in adaptive MCMC algorithms, ie mixing vs burning, with illustrations using the Wang-Landau algorithm.

Thus, as in the previous editions of Adap’ski, we have had a uniformly high quality of talks about the current research in the area of adaptive algorithms (and a wee further). This shows the field is very well active and expanding, aiming at reaching a wider audience by providing verifiable convergence conditions and semi-automated softwares (like Jeff Rosenthal’s amcmc R code we used in Introducing Monte Carlo Methods with R). Looking forward Adap’ski 4 (Adap’skiV?!), hopefully in Europe and why not in Chamonix?! Which could then lead us to call the next meeting Adap’skiX…