**A**part from subjecting my [surprisingly large!] audience to three hours of ABC tutorial today, and after running Ponte della la Libertà to Mestre and back in a deep fog, I attended the second part of the 1st Italian-French statistics seminar at Ca’Foscari, Venetiarum Universitas, with talks by Stéfano Tonellato and Roberto Casarin. Stéfano discussed a most interesting if puzzling notion of clustering via Dirichlet process mixtures. Which indeed puzzles me for its dependence on the Dirichlet measure and on the potential for an unlimited number of clusters as the sample size increases. The method offers similarities with an approach from our 2000 JASA paper on running inference on mixtures without proper label switching, in that looking at pairs of allocated observations to clusters is revealing about the [true or pseudo-true] number of clusters. With divergence in using eigenvalues of Laplacians on similarity matrices. But because of the potential for the number of components to diverge I wonder at the robustness of the approach via non-parametric [Bayesian] modelling. Maybe my difficulty stands with the very notion of cluster, which I find poorly defined and mostly in the eyes of the beholder! And Roberto presented a recent work on SURE and VAR models, with a great graphical representation of the estimated connections between factors in a sparse graphical model.

## Archive for clustering

## a Ca’Foscari [first Italian-French statistics seminar]

Posted in Kids, pictures, Statistics, Travel, University life with tags ABC, Approximate Bayesian computation, Ca' Foscari University, clustering, Dirichlet mixture priors, GRETA, Laplacian, mixtures, San Giobbe, tutorial, Università Italo-Francese, Université Franco-Italienne, Venezia, workshop on October 26, 2017 by xi'an## at CIRM [#2]

Posted in Mountains, pictures, Running, Statistics, Travel, University life with tags approximate MCMC, Cap Morgiou, CIRM, clustering, EM algorithm, functional programming languages, improper priors, Jeffreys-Lindley paradox, La Grande Candelle, Luminy, Marseille, Monte Carlo Statistical Methods, noisy MCMC, pGibbs, pMCMC, q-vague convergence, trail running, variable selection, yeast on March 2, 2016 by xi'an**S**ylvia Richardson gave a great talk yesterday on clustering applied to variable selection, which first raised [in me] a usual worry of the lack of background model for clustering. But the way she used this notion meant there was an infinite Dirichlet process mixture model behind. This is quite novel [at least for me!] in that it addresses the covariates and not the observations themselves. I still wonder at the meaning of the cluster as, if I understood properly, the dependent variable is not involved in the clustering. Check her R package PReMiuM for a practical implementation of the approach. Later, Adeline Samson showed us the results of using pMCM versus particle Gibbs for diffusion processes where (a) pMCMC was behaving much worse than particle Gibbs and (b) EM required very few particles and Metropolis-Hastings steps to achieve convergence, when compared with posterior approximations.

Today Pierre Druilhet explained to the audience of the summer school his measure theoretic approach [I discussed a while ago] to the limit of proper priors via *q-vague* convergence, with the paradoxical phenomenon that a Be(n⁻¹,n⁻¹) converges to a sum of two Dirac masses when the parameter space is [0,1] but to Haldane’s prior when the space is (0,1)! He also explained why the Jeffreys-Lindley paradox vanishes when considering different measures [with an illustration that came from my Statistica Sinica 1993 paper]. Pierre concluded with the above opposition between two Bayesian paradigms, a [sort of] tale of two sigma [fields]! Not that I necessarily agree with the first paradigm that priors are supposed to have generated the actual parameter. If only because it mechanistically excludes all improper priors…

Darren Wilkinson talked about yeast, which is orders of magnitude more exciting than it sounds, because this is Bayesian big data analysis in action! With significant (and hence impressive) results based on stochastic dynamic models. And massive variable selection techniques. Scala, Haskell, Frege, OCaml were [functional] languages he mentioned that I had never heard of before! And Daniel Rudolf concluded the [intense] second day of this Bayesian week at CIRM with a description of his convergence results for (rather controlled) noisy MCMC algorithms.

## mixtures of mixtures

Posted in pictures, Statistics, University life with tags arXiv, Austria, clustering, k-mean clustering algorithm, Linkz, map, MCMC, mixture, overfitting, Wien on March 9, 2015 by xi'an**A**nd yet another arXival of a paper on mixtures! This one is written by Gertraud Malsiner-Walli, Sylvia Frühwirth-Schnatter, and Bettina Grün, from the Johannes Kepler University Linz and the Wirtschaftsuniversitat Wien I visited last September. With the exact title being Identifying mixtures of mixtures using Bayesian estimation.

So, what *is* a mixture of mixtures if not a mixture?! Or if not *only* a mixture. The upper mixture level is associated with clusters, while the lower mixture level is used for modelling the distribution of a given cluster. Because the cluster needs to be real enough, the components of the mixture are assumed to be heavily overlapping. The paper thus spends a large amount of space on detailing the construction of the associated hierarchical prior. Which in particular implies defining through the prior what a cluster means. The paper also connects with the overfitting mixture idea of Rousseau and Mengersen (2011, Series B). At the cluster level, the Dirichlet hyperparameter is chosen to be very small, 0.001, which empties superfluous clusters but sounds rather arbitrary (which is the reason why we did not go for such small values in our testing/mixture modelling). On the opposite, the mixture weights have an hyperparameter staying (far) away from zero. The MCMC implementation is based on a standard Gibbs sampler and the outcome is analysed and sorted by estimating the “true” number of clusters as the MAP and by selecting MCMC simulations conditional on that value. From there clusters are identified via the point process representation of a mixture posterior. Using a standard k-means algorithm.

The remainder of the paper illustrates the approach on simulated and real datasets. Recovering in those small dimension setups the number of clusters used in the simulation or found in other studies. As noted in the conclusion, using solely a Gibbs sampler with such a large number of components is rather perilous since it may get stuck close to suboptimal configurations. Especially with very small Dirichlet hyperparameters.

## machine learning [book review]

Posted in Books, R, Statistics, University life with tags Bayesian statistics, clustering, data analysis, inference, machine learning, MAP estimators, MIT Press, statistics book on October 21, 2013 by xi'an**I** have to admit the rather embarrassing fact that *Machine Learning, A probabilistic perspective* by Kevin P. Murphy is the first machine learning book I really read in detail…! It is a massive book with close to 1,100 pages and I thus hesitated taking it with me around, until I grabbed it in my bag for Warwick. (And in the train to Argentan.) It is also massive in its contents as it covers most (all?) of what I call statistics (but visibly corresponds to machine learning as well!). With a Bayesian bent most of the time (which is the secret meaning of *probabilistic* in the title).

“…we define machine learning as a set of methods that can automatically detect patterns in data, and then use the uncovered patterns to predict future data, or to perform other kinds of decision making under uncertainty (such as planning how to collect more data!).” (p.1)

**A**part from the Introduction—which I find rather confusing for not dwelling on the nature of errors and randomness and on the reason for using probabilistic models (since they are all wrong) and charming for including a picture of the author’s family as an illustration of face recognition algorithms—, I cannot say I found the book more lacking in foundations or in the breadth of methods and concepts it covers than a “standard” statistics book. In short, this is a perfectly acceptable statistics book! Furthermore, it has a very relevant and comprehensive selection of references (sometimes favouring “machine learning” references over “statistics” references!). Even the vocabulary seems pretty standard to me. All this makes me wonder why we at all distinguish between the two domains, following Larry Wasserman’s views (for once!) that the difference is mostly in the eye of the beholder, i.e. in which department one teaches… Which was already my perspective before I read the book but it comforted me even further. And the author agrees as well *(“The probabilistic approach to machine learning is closely related to the field of statistics, but differs slightly in terms of its emphasis and terminology”, p.1).* Let us all unite!

[..part 2 of the book review to appear tomorrow…]

## AMOR at 5000ft in a water tank…

Posted in Mountains, pictures, Statistics, University life with tags adaptive MCMC, AMOR, Argentina, clustering, cosmic rays, label switching experiment, LAL, mixture estimation, mixtures, Oxford, permutations, Pierre-Augier experiment, relabelling, Université Paris-Sud, University of Oxford on November 22, 2012 by xi'an**O**n Monday, I attended the thesis defence of Rémi Bardenet in Orsay as a member (referee) of his thesis committee. While this was a thesis in computer science, which took place in the Linear Accelerator Lab in Orsay, it was clearly rooted in computational statistics, hence justifying my presence in the committee. The justification (!) for the splashy headline of this post is that Rémi’s work was motivated by the Pierre-Auger experiment on ultra-high-energy cosmic rays, where particles are detected through a network of 1600 water tanks spread over the Argentinian Pampa Amarilla on an area the size of Rhode Island (where I am incidentally going next week).

**T**he part of Rémi’s thesis presented during the defence concentrated on his AMOR algorithm, arXived in a paper written with Olivier Cappé and Gersende Fort. AMOR stands for adaptive Metropolis online relabelling and combines adaptive MCMC techniques with relabelling strategies to fight label-switching (e.g., in mixtures). I have been interested in mixtures for eons (starting in 1987 in Ottawa with applying Titterington, Smith, and Makov to chest radiographs) and in label switching for ages (starting at the COMPSTAT conférence in Bristol in 1998). Rémi’s approach to the label switching problem follows the relabelling path, namely a projection of the original parameter space into a smaller subspace (that is also a quotient space) to avoid permutation invariance and lack of identifiability. (In the survey I wrote with Kate Lee, Jean-Michel Marin and Kerrie Mengersen, we suggest using the mode as a pivot to determine which permutation to use on the components of the mixture.) The paper suggests using an Euclidean distance to a mean determined adaptively, μ_{t}, with a quadratic form Σ_{t} also determined on-the-go, minimising (Pθ-μ_{t})^{T}Σ_{t}(Pθ-μ_{t}) over all permutations P at each step of the algorithm. The intuition behind the method is that the posterior over the restricted space should look like a roughly elliptically symmetric distribution, or at least like a unimodal distribution, rather than borrowing bits and pieces from different modes. While I appreciate the technical *tour de force* represented by the proof of convergence of the AMOR algorithm, I remain somehow sceptical about the approach and voiced the following objections during the defence: first, the assumption that the posterior becomes unimodal under an appropriate restriction is not necessarily realistic. Secondary modes often pop in with real data (as in the counter-example we used in our paper with Alessandra Iacobucci and Jean-Michel Marin). Next, the whole apparatus of fighting multiple modes and non-identifiability, i.e. fighting label switching, is to fall back on posterior means as Bayes estimators. As stressed in our JASA paper with Gilles Celeux and Merrilee Hurn, there is no reason for doing so and there are several reasons for not doing so:

- it breaks down under model specification, i.e., when the number of components is not correct
- it does not improve the speed of convergence but, on the opposite, restricts the space visited by the Markov chain
- it may fall victim to the fatal attraction of secondary modes by fitting too small an ellipse around one of those modes
- it ultimately depends on the parameterisation of the model
- there is no reason for using posterior means in mixture problems, posterior modes or cluster centres can be used instead

I am therefore very much more in favour of producing a posterior distribution that is *as label switching as possible* (since the true posterior is completely symmetric in this respect). Post-processing the resulting sample can be done by using off-the-shelf clustering in the component space, derived from the point process representation used by Matthew Stephens in his thesis and subsequent papers. It also allows for a direct estimation of the number of components.

**I**n any case, this was a defence worth-attending that led me to think afresh about the label switching problem, with directions worth exploring next month while Kate Lee is visiting from Auckland. Rémi Bardenet is now headed for a postdoc in Oxford, a perfect location to discuss further label switching and to engage into new computational statistics research!

## workshop a Venezia (2)

Posted in pictures, Statistics, Travel, University life with tags ABC, approximate likelihood, Ca' Foscari University, clustering, composite likelihood, empirical likelihood, hypothesis testing, Italia, loss functions, normalising constant, point null hypotheses, Venezia on October 10, 2012 by xi'an**I** could only attend one day of the workshop on likelihood, approximate likelihood and nonparametric statistical techniques with some applications, and I wish I could have stayed a day longer (and definitely not only for the pleasure of being in Venezia!) Yesterday, Bruce Lindsay started the day with an extended review of composite likelihood, followed by recent applications of composite likelihood to clustering (I was completely unaware he had worked on the topic in the 80’s!). His talk was followed by several talks working on composite likelihood and other pseudo-likelihoods, which made me think about potential applications to ABC. During my tutorial talk on ABC, I got interesting questions on multiple testing and how to combine the different “optimal” summary statistics (*answer:* take all of them, it would not make sense to co;pare one pair with one summary statistic and another pair with another summary statistic), and on why we were using empirical likelihood rather than another pseudo-likelihood (*answer:* I do not have a definite answer. I guess it depends on the ease with which the pseudo-likelihood is derived and what we do with it. I would e.g. feel less confident to use the pairwise composite as a substitute likelihood rather than as the basis for a score function.) In the final afternoon, Monica Musio presented her joint work with Phil Dawid on score functions and their connection with pseudo-likelihood and estimating equations (another possible opening for ABC), mentioning a score family developped by Hyvärinen that involves the gradient of the square-root of a density, in the best James-Stein tradition! (Plus an approach bypassing the annoying missing normalising constant.) Then, based on a joint work with Nicola Satrori and Laura Ventura, Ruli Erlis exposed a 3rd-order tail approximation towards a (marginal) posterior simulation called HOTA. As Ruli will visit me in Paris in the coming weeks, I hope I can explore the possibilities of this method when he is (t)here. At last, Stéfano Cabras discussed higher-order approximations for Bayesian point-null hypotheses (jointly with Walter Racugno and Laura Ventura), mentioning the Pereira and Stern (so special) loss function mentioned in my post on Måns’ paper the very same day! It was thus a very informative and beneficial day for me, furthermore spent in a room overlooking the Canal Grande in the most superb location!