**O**ur paper Rethinking the Effective Sample Size, with Victor Elvira (the driving force behind the paper!) and Luca Martino, has now been published in the *International Statistical Review*! As discussed earlier on this blog, we wanted to re-evaluate the pros and cons of the effective sample size (ESS), as a tool assessing the quality [or lack thereof] of a Monte Carlo approximation. It is particularly exploited in the specific context of importance sampling. Following a 1992 construction by Augustine Kong, his approximation has been widely used in the last 25 years, in part due to its simplicity as a practical rule of thumb. However, we show in this paper that the assumptions made in the derivation of this approximation make it difficult to consider it as a reasonable approximation of the ESS. Note that this reevaluation does not cover the use of ESS for Markov chain Monte Carlo algorithms, although there would also be much to tell about it..!

## Archive for International Statistical Review

## rethinking the ESS published!

Posted in Statistics with tags effective sample size, ESS, importance sampling, International Statistical Review, Markov chain Monte Carlo algorithm, MCMC, Monte Carlo methods, Monte Carlo Statistical Methods, simulation on May 3, 2022 by xi'an## RB4MCMC@ISR

Posted in Statistics with tags 100th birthday, arXiv, C.R. Rao, International Statistical Review, Rao-Blackwell theorem, Rao-Blackwellisation, typos, University of Warwick on August 18, 2021 by xi'an**O**ur survey paper on Rao-Blackwellisation (and the first Robert&Roberts published paper!) just appeared on-line as part of the International Statistical Review mini-issue in honour of C.R. Rao on the occasion of his 100th birthday. (With an unfortunate omission of my affiliation with Warwick!). While the papers are unfortunately beyond a paywall, except for a few weeks!, the arXiv version is still available (and presumably with less typos!).

## Rao-Blackwellisation in the MCMC era

Posted in Books, Statistics, University life with tags auxiliary variables, birthday, C.R. Rao, conditioning, David Blackwell, demarginalisation, International Statistical Review, MCMC, Monte Carlo Statistical Methods, Rao-Blackwell theorem, Rao-Blackwellisation on January 6, 2021 by xi'an**A** few months ago, as indicated on this blog, I was contacted by ISR editors to write a piece on Rao-Blackwellisation, towards a special issue celebrating Calyampudi Radhakrishna Rao’s 100th birthday. Gareth Roberts and I came up with this survey, now on arXiv, discussing different aspects of Monte Carlo and Markov Chain Monte Carlo that pertained to Rao-Blackwellisation, one way or another. As I discussed the topic with several friends over the Fall, it appeared that the difficulty was more in setting the boundaries. Than in finding connections. In a way anything conditioning or demarginalising or resorting to auxiliary variates is a form of Rao-Blackwellisation. When re-reading the JASA Gelfand and Smith 1990 paper where I first saw the link between the Rao-Blackwell theorem and simulation, I realised my memory of it had drifted from the original, since the authors proposed there an approximation of the marginal based on replicas rather than the original Markov chain. Being much closer to Tanner and Wong (1987) than I thought. It is only later that the true notion took shape. *[Since the current version is still a draft, any comment or suggestion would be most welcomed!]*

## practical Bayesian inference [book review]

Posted in Books, Kids, R, Statistics, University life with tags ABC model choice, Bayesian hypothesis testing, Bayesian inference, book review, Cambridge University Press, International Statistical Review, Monte Carlo Statistical Methods, practical Bayesian inference, R, textbook on April 26, 2018 by xi'an*[Disclaimer: I received this book of Coryn Bailer-Jones for a review in the International Statistical Review and intend to submit a revised version of this post as my review. As usual, book reviews on the ‘Og are reflecting my own definitely personal and highly subjective views on the topic!]*

**I**t is always a bit of a challenge to review introductory textbooks as, on the one hand, they are rarely written at the level and with the focus one would personally choose to write them. And, on the other hand, it is all too easy to find issues with the material presented and the way it is presented… So be warned and proceed cautiously! In the current case, Practical Bayesian Inference tries to embrace too much, methinks, by starting from basic probability notions (that should not be unknown to physical scientists, I believe, and which would avoid introducing a flat measure as a uniform distribution over the real line!, p.20). All the way to running MCMC for parameter estimation, to compare models by Bayesian evidence, and to cover non-parametric regression and bootstrap resampling. For instance, priors only make their apparition on page 71. With a puzzling choice of an improper prior (?) leading to an improper posterior (??), which is certainly not the smoothest entry on the topic. “*Improper posteriors are a bad thing*“, indeed! And using truncation to turn them into proper distributions is not a clear improvement as the truncation point will significantly impact the inference. Discussing about the choice of priors from the beginning has some appeal, but it may also create confusion in the novice reader (although one never knows!). Even asking about “*what is a good prior?*” (p.73) is not necessarily the best (and my recommended) approach to a proper understanding of the Bayesian paradigm. And arguing about the unicity of the prior (p.119) clashes with my own view of the prior being primarily a reference measure rather than an ideal summary of the available information. (The book argues at some point that there is no fixed model parameter, another and connected source of disagreement.) There is a section on assigning priors (p.113), but it only covers the case of a possibly biased coin without much realism. A feature common to many Bayesian textbooks though. To return to the issue of improper priors (and posteriors), the book includes several warnings about the danger of hitting an undefined posterior (still called a distribution), without providing real guidance on checking for its definition. (A tough question, to be sure.)

“One big drawback of the Metropolis algorithm is that it uses a fixed step size, the magnitude of which can hardly be determined in advance…”(p.165)

When introducing computational techniques, quadratic (or Laplace) approximation of the likelihood is mingled with kernel estimators, which does not seem appropriate. Proposing to check convergence and calibrate MCMC via ACF graphs is helpful in low dimensions, but not in larger dimensions. And while warning about the dangers of forgetting the Jacobians in the Metropolis-Hastings acceptance probability when using a transform like η=ln θ is well-taken, the loose handling of changes of variables may be more confusing than helpful (p.167). Discussing and providing *two* R codes for the (standard) Metropolis algorithm may prove too much. Or not. But using a four page R code for fitting a simple linear regression with a flat prior (pp.182-186) may definitely put the reader off! Even though I deem the example a proper experiment in setting a Metropolis algorithm and appreciate the detailed description around the R code itself. (I just take exception at the paragraph on running the code with two or even one observation, as the fact that “*the Bayesian solution always exists*” (p.188) [under a proper prior] is not necessarily convincing…)

“In the real world we cannot falsify a hypothesis or model any more than we “truthify” it (…) All we can do is ask which of the available models explains the data best.” (p.224)

In a similar format, the discussion on testing of hypotheses starts with a lengthy presentation of classical tests and p-values, the chapter ending up with a list of issues. Most of them reasonable in my own referential. I also concur with the conclusive remarks quoted above that what matters is a comparison of (all relatively false) models. What I less agree [as predictable from earlier posts and papers] with is the (standard) notion that comparing two models with a Bayes factor follows from the no information (in order to avoid the heavily loaded non-informative) prior weights of ½ and ½. Or similarly that the evidence is uniquely calibrated. Or, again, using a truncated improper prior under one of the assumptions (with the ghost of the Jeffreys-Lindley paradox lurking nearby…). While the Savage-Dickey approximation is mentioned, the first numerical resolution of the approximation to the Bayes factor is via simulations from the priors. Which may be very poor in the situation of vague and uninformative priors. And then the deadly harmonic mean makes an entry (p.242), along with nested sampling… There is also a list of issues about Bayesian model comparison, including (strong) dependence on the prior, dependence on irrelevant alternatives, lack of goodness of fit tests, computational costs, including calls to possibly intractable likelihood function, ABC being then mentioned as a solution (which it is not, mostly).

## maximum likelihood: an introduction

Posted in Books, Statistics with tags Bahadur, International Statistical Review, Likelihood Principle, Lucien Le Cam, maximum likelihood estimation on December 20, 2014 by xi'an

“Basic Principle 0. Do not trust any principle.” L. Le Cam (1990)

**H**ere is the abstract of a International Statistical Rewiew 1990 paper by Lucien Le Cam on maximum likelihood. ISR keeping a tradition of including an abstract in French for every paper, Le Cam (most presumably) wrote his own translation [or maybe wrote the French version first], which sounds much funnier to me and so I cannot resist posting both, pardon my/his French! *[I just find “Ce fait” rather unusual, as I would have rather written “Ceci fait”…]*:

Maximum likelihood estimates are reported to be best under all circumstances. Yet there are numerous simple examples where they plainly misbehave. One gives some examples for problems that had not been invented for the purpose of annoying maximum likelihood fans. Another example, imitated from Bahadur, has been specially created with just such a purpose in mind. Next, we present a list of principles leading to the construction of good estimates. The main principle says that one should not believe in principles but study each problem for its own sake.

L’auteur a ouï dire que la méthode du maximum de vraisemblance est la meilleure méthode d’estimation. C’est bien vrai, et pourtant la méthode se casse le nez sur des exemples bien simples qui n’avaient pas été inventés pour le plaisir de montrer que la méthode peut être très désagréable. On en donne quelques-uns, plus un autre, imité de Bahadur et fabriqué exprès pour ennuyer les admirateurs du maximum de vraisemblance. Ce fait, on donne une savante liste de principes de construction de bons estimateurs, le principe principal étant qu’il ne faut pas croire aux principes.

The entire paper is just as witty, as in describing the mixture model as “contaminated and not fit to drink”! Or in “Everybody knows that taking logarithms is unfair”. Or, again, in “biostatisticians, being complicated people, prefer to work out not with the dose y but with its logarithm”… And a last line: “One possibility is that there are too many horse hairs in e”.