Archive for clustering

the most probable cluster

Posted in Books, Statistics with tags , , , , , , on July 11, 2019 by xi'an

In the last issue of Bayesian Analysis, Lukasz Rajkowski studies the most likely (MAP) cluster associated with the Dirichlet process mixture model. Reminding me that most Bayesian estimates of the number of clusters are not consistent (when the sample size grows to infinity). I am always puzzled by this problem, as estimating the number of clusters sounds like an ill-posed problem, since it is growing with the number of observations, by definition of the Dirichlet process. For instance, the current paper establishes that the number of clusters intersecting a given compact set remains bounded. (The setup is one of a Normal Dirichlet process mixture with constant and known covariance matrix.)

Since the posterior probability of a given partition of {1,2,…,n} can be (formally) computed, the MAP estimate can be (formally) derived. I inserted formally in the previous sentence as the derivation of the exact MAP is an NP hard problem in the number n of observations. As an aside, I have trouble with the author’s argument that the convex hulls of the clusters should be disjoin: I do not see why they should when the mixture components are overlapping. (More generally, I fail to relate to notions like “bad clusters” or “overestimation of the number of clusters” or a “sensible choice” of the covariance matrix.) More globally, I am somewhat perplexed by the purpose of the paper and the relevance of the MAP estimate, even putting aside my generic criticisms of the MAP approach. No uncertainty is attached to the estimator, which thus appears as a form of penalised likelihood strategy rather than a genuinely Bayesian (Analysis) solution.

The first example in the paper is using data from a Uniform over (-1,1), concluding at a “misleading” partition by the MAP since it produces more than one cluster. I find this statement flabbergasting as the generative model is not the estimated model. To wit, the case of an exponential Exp(1) sample that cannot reach a maximum of the target function with a finite number of sample. Which brings me back full-circle to my general unease about clustering in that much more seems to be assumed about this notion than what the statistical model delivers.

O’Bayes 19/1 [snapshots]

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , , , , , , , on June 30, 2019 by xi'an

Although the tutorials of O’Bayes 2019 of yesterday were poorly attended, albeit them being great entries into objective Bayesian model choice, recent advances in MCMC methodology, and the multiple layers of BART, for which I have to blame myself for sticking the beginning of O’Bayes too closely to the end of BNP as only the most dedicated could achieve the commuting from Oxford to Coventry to reach Warwick in time, the first day of talks were well attended, despite weekend commitments, conference fatigue, and perfect summer weather! Here are some snapshots from my bench (and apologies for not covering better the more theoretical talks I had trouble to follow, due to an early and intense morning swimming lesson! Like Steve Walker’s utility based derivation of priors that generalise maximum entropy priors. But being entirely independent from the model does not sound to me like such a desirable feature… And Natalia Bochkina’s Bernstein-von Mises theorem for a location scale semi-parametric model, including a clever construct of a mixture of two Dirichlet priors to achieve proper convergence.)

Jim Berger started the day with a talk on imprecise probabilities, involving the society for imprecise probability, which I discovered while reading Keynes’ book, with a neat resolution of the Jeffreys-Lindley paradox, when re-expressing the null as an imprecise null, with the posterior of the null no longer converging to one, with a limit depending on the prior modelling, if involving a prior on the bias as well, with Chris discussing the talk and mentioning a recent work with Edwin Fong on reinterpreting marginal likelihood as exhaustive X validation, summing over all possible subsets of the data [using log marginal predictive].Håvard Rue did a follow-up talk from his Valencià O’Bayes 2015 talk on PC-priors. With a pretty hilarious introduction on his difficulties with constructing priors and counseling students about their Bayesian modelling. With a list of principles and desiderata to define a reference prior. However, I somewhat disagree with his argument that the Kullback-Leibler distance from the simpler (base) model cannot be scaled, as it is essentially a log-likelihood. And it feels like multivariate parameters need some sort of separability to define distance(s) to the base model since the distance somewhat summarises the whole departure from the simpler model. (Håvard also joined my achievement of putting an ostrich in a slide!) In his discussion, Robin Ryder made a very pragmatic recap on the difficulties with constructing priors. And pointing out a natural link with ABC (which brings us back to Don Rubin’s motivation for introducing the algorithm as a formal thought experiment).

Sara Wade gave the final talk on the day about her work on Bayesian cluster analysis. Which discussion in Bayesian Analysis I alas missed. Cluster estimation, as mentioned frequently on this blog, is a rather frustrating challenge despite the simple formulation of the problem. (And I will not mention Larry’s tequila analogy!) The current approach is based on loss functions directly addressing the clustering aspect, integrating out the parameters. Which produces the interesting notion of neighbourhoods of partitions and hence credible balls in the space of partitions. It still remains unclear to me that cluster estimation is at all achievable, since the partition space explodes with the sample size and hence makes the most probable cluster more and more unlikely in that space. Somewhat paradoxically, the paper concludes that estimating the cluster produces a more reliable estimator on the number of clusters than looking at the marginal distribution on this number. In her discussion, Clara Grazian also pointed the ambivalent use of clustering, where the intended meaning somehow diverges from the meaning induced by the mixture model.

a book and two chapters on mixtures

Posted in Books, Statistics, University life with tags , , , , , , , , , , , , , on January 8, 2019 by xi'an

The Handbook of Mixture Analysis is now out! After a few years of planning, contacts, meetings, discussions about notations, interactions with authors, further interactions with late authors, repeating editing towards homogenisation, and a final professional edit last summer, this collection of nineteen chapters involved thirty-five contributors. I am grateful to all participants to this piece of work, especially to Sylvia Früwirth-Schnatter for being a driving force in the project and for achieving a much higher degree of homogeneity in the book than I expected. I would also like to thank Rob Calver and Lara Spieker of CRC Press for their boundless patience through the many missed deadlines and their overall support.

Two chapters which I co-authored are now available as arXived documents:

5. Gilles Celeux, Kaniav Kamary, Gertraud Malsiner-Walli, Jean-Michel Marin, and Christian P. Robert, Computational Solutions for Bayesian Inference in Mixture Models
7. Gilles Celeux, Sylvia Früwirth-Schnatter, and Christian P. Robert, Model Selection for Mixture Models – Perspectives and Strategies

along other chapters

1. Peter Green, Introduction to Finite Mixtures
8. Bettina Grün, Model-based Clustering
12. Isobel Claire Gormley and Sylvia Früwirth-Schnatter, Mixtures of Experts Models
13. Sylvia Kaufmann, Hidden Markov Models in Time Series, with Applications in Economics
14. Elisabeth Gassiat, Mixtures of Nonparametric Components and Hidden Markov Models
19. Michael A. Kuhn and Eric D. Feigelson, Applications in Astronomy

Big Bayes goes South

Posted in Books, Mountains, pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , , , , , , on December 5, 2018 by xi'an

At the Big [Data] Bayes conference this week [which I found quite exciting despite a few last minute cancellations by speakers] there were a lot of clustering talks including the ones by Amy Herring (Duke), using a notion of centering that should soon appear on arXiv. By Peter Müller (UT, Austin) towards handling large datasets. Based on a predictive recursion that takes one value at a time, unsurprisingly similar to the update of Dirichlet process mixtures. (Inspired by a 1998 paper by Michael Newton and co-authors.) The recursion doubles in size at each observation, requiring culling of negligible components. Order matters? Links with Malsiner-Walli et al. (2017) mixtures of mixtures. Also talks by Antonio Lijoi and Igor Pruenster (Boconni Milano) on completely random measures that are used in creating clusters. And by Sylvia Frühwirth-Schnatter (WU Wien) on creating clusters for the Austrian labor market of the impact of company closure. And by Gregor Kastner (WU Wien) on multivariate factor stochastic models, with a video of a large covariance matrix evolving over time and catching economic crises. And by David Dunson (Duke) on distance clustering. Reflecting like myself on the definitely ill-defined nature of the [clustering] object. As the sample size increases, spurious clusters appear. (Which reminded me of a disagreement I had had with David McKay at an ICMS conference on mixtures twenty years ago.) Making me realise I missed the recent JASA paper by Miller and Dunson on that perspective.

Some further snapshots (with short comments visible by hovering on the picture) of a very high quality meeting [says one of the organisers!]. Following suggestions from several participants, it would be great to hold another meeting at CIRM in a near future. Continue reading

Handbook of Mixture Analysis [cover]

Posted in Books, Statistics, University life with tags , , , , , , , , on August 15, 2018 by xi'an

On the occasion of my talk at JSM2018, CRC Press sent me the cover of our incoming handbook on mixture analysis, courtesy of Rob Calver who managed to get it to me on very short notice! We are about ready to send the manuscript to CRC Press and hopefully the volume will get published pretty soon. It would have been better to have it ready for JSM2018, but we editors got delayed by a few months for the usual reasons.

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

Posted in Kids, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , on October 26, 2017 by xi'an

Apart 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.

at CIRM [#2]

Posted in Mountains, pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , , on March 2, 2016 by xi'an

Sylvia 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.