**O**ne of my students in my MCMC course at ENSAE seems to specialise into spotting typos in the Monte Carlo Statistical Methods book as he found an issue in every problem he solved! He even went back to a 1991 paper of mine on Inverse Normal distributions, inspired from a discussion with an astronomer, Caroline Soubiran, and my two colleagues, Gilles Celeux and Jean Diebolt. The above derivation from the massive Gradsteyn and Ryzhik (which I discovered thanks to Mary Ellen Bock when arriving in Purdue) is indeed incorrect as the final term should be the square root of 2β rather than 8β. However, this typo does not impact the normalising constant of the density, K(α,μ,τ), unless I am further confused.

## Archive for MCMC

## I thought I did make a mistake but I was wrong…

Posted in Books, Kids, Statistics with tags Charles M. Schulz, confluent hypergeometric function, course, ENSAE, exercises, Gradsteyn, inverse normal distribution, MCMC, mixtures, Monte Carlo Statistical Methods, Peanuts, Ryzhik, typos on November 14, 2018 by xi'an## rethinking the ESS

Posted in Statistics with tags arXiv, delta method, effective sample size, efficiency measures, efficient importance sampling, ESS, importance sampling, MCMC, Monte Carlo Statistical Methods, simulation on September 14, 2018 by xi'an**F**ollowing Victor Elvira‘s visit to Dauphine, one and a half year ago, where we discussed the many defects of ESS as a default measure of efficiency for importance sampling estimators, and then some more efforts (mostly from Victor!) to formalise these criticisms, Victor, Luca Martino and I wrote a paper on this notion, now arXived. (Victor most kindly attributes the origin of the paper to a 2010 ‘Og post on the topic!) The starting thread of the (re?)analysis of this tool introduced by Kong (1992) is that the ESS used in the literature is an *approximation* to the “true” ESS, generally unavailable. Approximation that is pretty crude and hence impacts the relevance of using it as *the* assessment tool for comparing importance sampling methods. In the paper, we re-derive (with the uttermost precision) the resulting approximation and list the many assumptions that [would] validate this approximation. The resulting drawbacks are many, from the absurd property of always being worse than direct sampling, to being independent from the target function and from the sample *per se*. Since only importance weights matter. This list of issues is not exactly brand new, but we think it is worth signaling given the fact that this approximation has been widely used in the last 25 years, due to its simplicity, as a practical rule of thumb [!] in a wide variety of importance sampling methods. In continuation of the directions drafted in Martino et al. (2017), we also indicate some alternative notions of importance efficiency. Note that this paper does not cover the use of ESS for MCMC algorithms, where it is somewhat more legit, if still too rudimentary to really catch convergence or lack thereof! *[Note: I refrained from the post title resinking the ESS…]*

## off to Singapore [IMS workshop]

Posted in pictures, Statistics, Travel, University life with tags ABC, IMS, Institute of Mathematical Statistics, MCMC, multilevel Monte Carlo, NUS, particle filters, Singapore, workshop on August 26, 2018 by xi'an**T**onight I am off to the National University of Singapore, at the Institute for Mathematical Sciences [and not the Institute of Mathematical Statistics!], to take part in a (first) week workshop on Bayesian Computation for High-Dimensional Statistical Models, covering topics like Approximate Bayesian Computation, Markov chain Monte Carlo, Multilevel Monte Carlo and Particle Filters. Having just barely recovered from the time difference with Vancouver, I now hope I can switch with not too much difficulty to Singapore time zone! As well as face the twenty plus temperature gap with the cool weather this morning in the Parc…

## JSM 2018 [#3]

Posted in Mountains, pictures, Statistics, Travel, University life with tags British Columbia, Canada, COPSS Award, curse of dimensionality, Grouse Mountain, HMC, JSM 2018, MCMC, Monte Carlo Statistical Methods, Vancouver, waterplane on August 2, 2018 by xi'an**T**hird day at JSM2018 and the audience is already much smaller than the previous days! Although it is hard to tell with a humongous conference centre spread between two buildings. And not getting hooked by the tantalising view of the bay, with waterplanes taking off every few minutes…

Still, there were (too) few participants in the two computational statistics (MCMC) sessions I attended in the morning, the first one being organised by James Flegal on different assessments of MCMC convergence. (Although this small audience made the session quite homely!) In his own talk, James developed an interesting version of multivariate ESS that he related with a stopping rule for minimal precision. Vivek Roy also spoke about a multiple importance sampling construction I missed when it came upon on arXiv last May.

In the second session, Mylène Bédard exposed the construction of and improvement brought by local scaling in MALA, with 20% gain from using non-local tuning. Making me idle muse over whether block sizes in block-Gibbs sampling could also be locally optimised… Then Aaron Smith discussed how HMC should be scaled for optimal performances, under rather idealised conditions and very high dimensions. Mentioning a running time of d, the dimension, to the power ¼. But not addressing the practical question of calibrating scale versus number of steps in the discretised version. (At which time my hands were [sort of] frozen solid thanks to the absurd air conditioning in the conference centre and I had to get out!)

## independent random sampling methods [book review]

Posted in Books, Statistics, University life with tags book review, inverse cdf, MCMC, Monte Carlo methods, Monte Carlo Statistical Methods, multiple try Metropolis, Non-Uniform Random Variate Generation, PRNG, random number generation, ratio of uniform algorithm, simulation, Springer-Verlag, Universidad Carlos III de Madrid, vertical density representation on May 16, 2018 by xi'an**L**ast week, I had the pleasant surprise to receive a copy of this book in the mail. Book that I was not aware had been written or published (meaning that I was not involved in its review!). The three authors, Luca Martino, David Luengo, and Joaquín Míguez, of Independent Random Sampling Methods are from Madrid universities and I have read (and posted on) several of their papers on (population) Monte Carlo simulation in the recent years. Including Luca’s survey of multiple try MCMC which was helpful in writing our WIREs own survey.

The book is a pedagogical coverage of most algorithms used to simulate independent samples from a given distribution, which of course recoups some of the techniques exposed with more details by [another] Luc, namely Luc Devroye’s Non-uniform random variate generation bible, often mentioned here (and studied in uttermost details by a dedicated reading group in Warwick). It includes a whole chapter on accept-reject methods, with in particular a section on Payne-Dagpunar’s band rejection I had not seen previously. And another entire chapter on ratio-of-uniforms techniques. On which the three authors had proposed generalisations [covered by the book], years before I attempted to go the same way, having completely forgotten reading their paper at the time… Or the much earlier 1991 paper by Jon Wakefield, Alan Gelfand and Adrian Smith!

The book also covers the “vertical density representation”, due to Troutt (1991), which consists in considering the distribution of the density p(.) of the random variable X as a random variable, p(X). I remember pondering about this alternative to the cdf transform and giving up on it as the outcome has a distribution depending on p, even when the density is monotonous. Even though I am not certain from reading the section that this is particularly appealing…

Given its title, the book contains very little about MCMC. Except for a last and final chapter that covers adaptive independent Metropolis-Hastings algorithms, in connection with some of the authors’ recent work. Like multiple try Metropolis. Relating to the (unidimensional) ARMS “ancestor” of adaptive MCMC methods. (As noted in a recent blog on Holden et al., 2009 , I have trouble understanding how recycling only rejected proposed values to build a better proposal distribution is enough to guarantee convergence of an adaptive algorithm, but the book does not delve much into this convergence.)

All in all and with the bias induced by me working in the very area, I find the book quite a nice entry on the topic, which can be used in a Monte Carlo course at both undergraduate and graduate levels if one want to avoid going into Markov chains. It is certainly less likely to scare students away than the comprehensive Non-uniform random variate generation and on the opposite may induce some of them to pursue a research career in this domain.

## accelerating MCMC

Posted in Books, Statistics, University life with tags acceleration of MCMC algorithms, algorithms, arXiv, cross validated, MCMC, Monte Carlo Statistical Methods, referee, simulation, Telecom Lille, typology, Université Paris Dauphine, University of Warwick, WIREs on April 11, 2018 by xi'an**A**s forecasted a rather long while ago (!), I wrote a short and incomplete survey on some approaches to accelerating MCMC. With the massive help of Victor Elvira (Lille), Nick Tawn (Warwick) and Changye Wu (Dauphine). Survey which current version just got arXived and which has now been accepted by WIREs Computational Statistics. The typology (and even the range of methods) adopted here is certainly mostly arbitrary, with suggestions for different divisions made by a very involved and helpful reviewer. While we achieved a quick conclusion to the review process, suggestions and comments are most welcome! Even if we cannot include every possible suggestion, just like those already made on X validated. (WIREs stands for Wiley Interdisciplinary Reviews and its dozen topics cover several fields, from computational stats to biology, to medicine, to engineering.)