**S**ophie Donnet pointed out to me this arXived paper by Tianxi Li, Elizaveta Levina, and Ji Zhu, on a network resampling strategy for X validation, where I appear as a datapoint rather than as a [direct] citation! Which reminded me of the “where you are the hero” gamebooks with which my kids briefly played, before computer games took over. The model selection method is illustrated on a dataset made of X citations [reduced to 706 authors] in all papers published between 2003 and 2012 in the Annals of Statistics, Biometrika, JASA, and JRSS Series B. With the outcome being the determination of a number of communities, 20, which the authors labelled as they wanted, based on 10 authors with the largest number of citations in the category. As it happens, I appear in the list, within the “mixed (causality + theory + Bayesian)” category (!), along with Jamie Robbins, Paul Fearnhead, Gilles Blanchard, Zhiqiang Tan, Stijn Vansteelandt, Nancy Reid, Jae Kwang Kim, Tyler VanderWeele, and Scott Sisson, which is somewhat mind-boggling in that I am pretty sure I never quoted six of these authors [although I find it hilarious that Jamie appears in the category, given that we almost got into a car crash together, at one of the Valencià meetings!].

## Archive for JASA

## Bayesian inference with intractable normalizing functions

Posted in Books, Statistics with tags adaptive MCMC methods, American Statistical Association, auxiliary variable, benchmark, doubly intractable problems, importance sampling, Ising model, JASA, MCMC algorithms, noisy MCMC, normalising constant, Russian roulette on December 13, 2018 by xi'an**I**n the latest September issue of JASA I received a few days ago, I spotted a review paper by Jaewoo Park & Murali Haran on intractable normalising constants Z(θ). There have been many proposals for solving this problem as well as several surveys, some conferences and even a book. The current survey focus on MCMC solutions, from auxiliary variable approaches to likelihood approximation algorithms (albeit without ABC entries, even though the 2006 auxiliary variable solutions of Møller et al. et of Murray et al. do simulate pseudo-observations and hence…). This includes the MCMC approximations to auxiliary sampling proposed by Faming Liang and co-authors across several papers. And the paper Yves Atchadé, Nicolas Lartillot and I wrote ten years ago on an adaptive MCMC targeting Z(θ) and using stochastic approximation à la Wang-Landau. Park & Haran stress the relevance of using sufficient statistics in this approach towards fighting computational costs, which makes me wonder if an ABC version could be envisioned. The paper also includes pseudo-marginal techniques like Russian Roulette (once spelled Roullette) and noisy MCMC as proposed in Alquier et al. (2016). These methods are compared on three examples: (1) the Ising model, (2) a social network model, the Florentine business dataset used in our original paper, and a larger one where most methods prove too costly, and (3) an attraction-repulsion point process model. In conclusion, an interesting survey, taking care to spell out the calibration requirements and the theoretical validation, if of course depending on the chosen benchmarks.

## empirical Bayes, reference priors, entropy & EM

Posted in Mountains, Statistics, Travel, University life with tags arXiv, Darjeeling, EM algorithm, empirical Bayes, I.J. Good, JASA, Kullback-Leibler divergence, MLE, non-parametrics, penalty, reparameterisation, Robbins-Monro algorithm on January 9, 2017 by xi'an**K**lebanov and co-authors from Berlin arXived this paper a few weeks ago and it took me a quiet evening in Darjeeling to read it. It starts with the premises that led Robbins to introduce empirical Bayes in 1956 (although the paper does not appear in the references), where repeated experiments with different parameters are run. Except that it turns non-parametric in estimating the prior. And to avoid resorting to the non-parametric MLE, which is the empirical distribution, it adds a smoothness penalty function to the picture. (**Warning:** I am not a big fan of non-parametric MLE!) The idea seems to have been Good’s, who acknowledged using the entropy as penalty is missing in terms of reparameterisation invariance. Hence the authors suggest instead to use as penalty function on the prior a joint relative entropy on both the parameter and the prior, which amounts to the average of the Kullback-Leibler divergence between the sampling distribution and the predictive based on the prior. Which is then independent of the parameterisation. And of the dominating measure. This is the only tangible connection with *reference priors* found in the paper.

The authors then introduce a non-parametric EM algorithm, where the unknown prior becomes the “parameter” and the M step means optimising an entropy in terms of this prior. With an infinite amount of data, the true prior (meaning the overall distribution of the genuine parameters in this repeated experiment framework) is a fixed point of the algorithm. However, it seems that the only way it can be implemented is via discretisation of the parameter space, which opens a whole Pandora box of issues, from discretisation size to dimensionality problems. And to motivating the approach by regularisation arguments, since the final product remains an atomic distribution.

While the alternative of estimating the marginal density of the data by kernels and then aiming at the closest entropy prior is discussed, I find it surprising that the paper does not consider the rather natural of setting a prior on the prior, e.g. via Dirichlet processes.

## male masters

Posted in Running, Travel with tags California, Golden Gate Bridge, Grand Master, half-marathon, JASA, plaque, podium, road running, San Francisco half-marathon on November 5, 2016 by xi'an**Y**esterday, I received this wood plaque from the organisers of the San Francisco Marathon! Nice, as I had not expected them to deliver abroad. And with this surprising title of second place in the male masters. After checking on Google (and avoiding a long list of SM sites), it appears that this means the above 40 category, which I had never heard of before. (I would have preferred an above 50 category, obviously! Which appears to exist in some races as the Grand Master category, even better!) Looking further at some of the (running) links, it also appears that categories by weight have been introduced in some races… Not that I can expect to benefit from that further division!

## adaptive exchange

Posted in Books, Statistics, University life with tags adaptive MCMC methods, auxiliary variables, bias, doubly intractable problems, evolutionary Monte Carlo, JASA, Markov chain Monte Carlo algorithm, Monte Carlo Statistical Methods, normalising constant, perfect sampling, simulated annealing on October 27, 2016 by xi'an**I**n the March 2016 issue of JASA that currently sits on my desk, there is a paper by Liang, Jim, Song and Liu on the adaptive exchange algorithm, which aims at handling posteriors for sampling distributions with intractable normalising constants. The concept behind the algorithm is the exchange principle initiated by Jesper Møller and co-authors in 2006, where an auxiliary pseudo-observation is simulated for the missing constants to vanish in a Metropolis-Hastings ratio. (The name *exchangeable* was introduced in a subsequent paper by Iain Murray, Zoubin Ghahramani and David MacKay, also in 2006.)

The crux of the method is to run an iteration as [where y denotes the observation]

- Proposing a new value θ’ of the parameter from a proposal q(θ’|θ);
- Generate a pseudo-observation z~ƒ(z|θ’);
- Accept with probability

which has the appeal to cancel all normalising constants. And the repeal of requiring an *exact* simulation from the very distribution with the missing constant, ƒ(.|θ). Which means that in practice a *finite* number of MCMC steps will be used and will *bias* the outcome. The algorithm is unusual in that it replaces the exact proposal q(θ’|θ) with an unbiased random version q(θ’|θ)ƒ(z|θ’), z being just an augmentation of the proposal. (The current JASA paper by Liang et al. seems to confuse *augment* and *argument*, see p.378.)

To avoid the difficulty in simulating from ƒ(.|θ), the authors draw pseudo-observations from sampling distributions with a *finite* number m of parameter values under the [unrealistic] assumption (A⁰) that this collection of values provides an almost complete cover of the posterior support. One of the tricks stands with an auxiliary [time-heterogeneous] chain of pseudo-observations generated by single Metropolis steps from one of these m fixed targets. These pseudo-observations are then used in the main (or *target*) chain to define the above exchange probability. The auxiliary chain is Markov but time-heterogeneous since the probabilities of accepting a move are evolving with time according to a simulated annealing schedule. Which produces a convergent estimate of the m normalising constants. The main chain is not Markov in that it depends on the whole history of the auxiliary chain [see Step 5, p.380]. Even jointly the collection of both chains is not Markov. The paper prefers to consider the process as an adaptive Markov chain. I did not check the rather intricate in details, so cannot judge of the validity of the overall algorithm; I simply note that one condition (A², p.383) is incredibly strong in that it assumes the Markov transition kernel to be Doeblin uniformly on any compact set of the calibration parameters. However, the major difficulty with this approach seems to be in its delicate calibration. From providing a reference set of m parameter values scanning the posterior support to picking transition kernels on both the parameter and the sample spaces, to properly cooling the annealing schedule [always a fun part!], there seems to be [from my armchair expert’s perspective, of course!] a wide range of opportunities for missing the target or running into zero acceptance problems. Both examples analysed in the paper, the auto-logistic and the auto-normal models, are actually of limited complexity in that they depend on a few parameters, 2 and 4 resp., and enjoy sufficient statistics, of dimensions 2 and 4 as well. Hence simulating (pseudo-)realisations of those sufficient statistics should be less challenging than the original approach replicating an entire vector of thousands of dimensions.

## finite mixture models [book review]

Posted in Books, Kids, Statistics, University life with tags Bayesian inference, David Peel, EM algorithm, finite mixtures, Geoff McLachlan, hidden Markov models, JASA, Markov switching models, MCMC, mixture estimation, Monte Carlo Statistical Methods, SAEM on February 17, 2014 by xi'an**H**ere is a review of Finite Mixture Models (2000) by Geoff McLachlan & David Peel that I wrote aeons ago (circa 1999), supposedly for JASA, which lost first the files and second the will to publish it. As I was working with my student today, I mentioned the book to her and decided to publish it here, if only because I think the book deserved a positive review, even after all those years! (Since then, Sylvia Frühwirth-Schnatter published Finite Mixture and Markov Switching Models (2004), which is closer to my perspective on the topic and that I would more naturally recommend.)

Mixture modeling, that is, the use of weighted sums of standard distributions as in

is a widespread and increasingly used technique to overcome the rigidity of standard parametric distributions such as f(y;**θ)**, while retaining a parametric nature, as exposed in the introduction of my JASA review to Böhning’s (1998) book on non-parametric mixture estimation (Robert, 2000). This review pointed out that, while there are many books available on the topic of mixture estimation, the unsurpassed reference remained the book by Titterington, Smith and Makov (1985) [hereafter TSM]. I also suggested that a new edition of TSM would be quite timely, given the methodological and computational advances that took place in the past 15 years: while it remains unclear whether or not this new edition will ever take place, the book by McLachlan and Peel gives an enjoyable and fairly exhaustive update on the topic, incorporating the most recent advances on mixtures and some related models.

Geoff McLachlan has been a major actor in the field for at least 25 years, through papers, software—the book concludes with a review of existing software—and books: McLachlan (1992), McLachlan and Basford (1988), and McLachlan and Krishnan (1997). I refer the reader to Lindsay (1989) for a review of the second book, which is a forerunner of, and has much in common with, the present book. Continue reading