Archive for unknown number of components

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

repulsive mixtures

Posted in Books, Statistics with tags , , , , , , , , on April 10, 2017 by xi'an

Fangzheng Xie and Yanxun Xu arXived today a paper on Bayesian repulsive modelling for mixtures. Not that Bayesian modelling is repulsive in any psychological sense, but rather that the components of the mixture are repulsive one against another. The device towards this repulsiveness is to add a penalty term to the original prior such that close means are penalised. (In the spirit of the sugar loaf with water drops represented on the cover of Bayesian Choice that we used in our pinball sampler, repulsiveness being there on the particles of a simulated sample and not on components.) Which means a prior assumption that close covariance matrices are of lesser importance. An interrogation I have has is was why empty components are not excluded as well, but this does not make too much sense in the Dirichlet process formulation of the current paper. And in the finite mixture version the Dirichlet prior on the weights has coefficients less than one.

The paper establishes consistency results for such repulsive priors, both for estimating the distribution itself and the number of components, K, under a collection of assumptions on the distribution, prior, and repulsiveness factors. While I have no mathematical issue with such results, I always wonder at their relevance for a given finite sample from a finite mixture in that they give an impression that the number of components is a perfectly estimable quantity, which it is not (in my opinion!) because of the fluid nature of mixture components and therefore the inevitable impact of prior modelling. (As Larry Wasserman would pound in, mixtures like tequila are evil and should likewise be avoided!)

The implementation of this modelling goes through a “block-collapsed” Gibbs sampler that exploits the latent variable representation (as in our early mixture paper with Jean Diebolt). Which includes the Old Faithful data as an illustration (for which a submission of ours was recently rejected for using too old datasets). And use the logarithm of the conditional predictive ordinate as  an assessment tool, which is a posterior predictive estimated by MCMC, using the data a second time for the fit.

Dirichlet process mixture inconsistency

Posted in Books, Statistics with tags , , , , on February 15, 2016 by xi'an

cover of Mixture Estimation and ApplicationsJudith Rousseau pointed out to me this NIPS paper by Jeff Miller and Matthew Harrison on the possible inconsistency of Dirichlet mixtures priors for estimating the (true) number of components in a (true) mixture model. The resulting posterior on the number of components does not concentrate on the right number of components. Which is not the case when setting a prior on the unknown number of components of a mixture, where consistency occurs. (The inconsistency results established in the paper are actually focussed on iid Gaussian observations, for which the estimated number of Gaussian components is almost never equal to 1.) In a more recent arXiv paper, they also show that a Dirichlet prior on the weights and a prior on the number of components can still produce the same features as a Dirichlet mixtures priors. Even the stick breaking representation! (Paper that I already reviewed last Spring.)

Bruce Lindsay (March 7, 1947 — May 5, 2015)

Posted in Books, Running, Statistics, Travel, University life with tags , , , , , , , , , , , on May 22, 2015 by xi'an

When early registering for Seattle (JSM 2015) today, I discovered on the ASA webpage the very sad news that Bruce Lindsay had passed away on May 5.  While Bruce was not a very close friend, we had met and interacted enough times for me to feel quite strongly about his most untimely death. Bruce was indeed “Mister mixtures” in many ways and I have always admired the unusual and innovative ways he had found for analysing mixtures. Including algebraic ones through the rank of associated matrices. Which is why I first met him—besides a few words at the 1989 Gertrude Cox (first) scholarship race in Washington DC—at the workshop I organised with Gilles Celeux and Mike West in Aussois, French Alps, in 1995. After this meeting, we met twice in Edinburgh at ICMS workshops on mixtures, organised with Mike Titterington. I remember sitting next to Bruce at one workshop dinner (at Blonde) and him talking about his childhood in Oregon and his father being a journalist and how this induced him to become an academic. He also contributed a chapter on estimating the number of components [of a mixture] to the Wiley book we edited out of this workshop. Obviously, his work extended beyond mixtures to a general neo-Fisherian theory of likelihood inference. (Bruce was certainly not a Bayesian!) Last time, I met him, it was in Italia, at a likelihood workshop in Venezia, October 2012, mixing Bayesian nonparametrics, intractable likelihoods, and pseudo-likelihoods. He gave a survey talk about composite likelihood, telling me about his extended stay in Italy (Padua?) around that time… So, Bruce, I hope you are now running great marathons in a place so full of mixtures that you can always keep ahead of the pack! Fare well!

 

Overfitting Bayesian mixture models with an unknown number of components

Posted in Statistics with tags , , , , , , , , on March 4, 2015 by xi'an

During my Czech vacations, Zoé van Havre, Nicole White, Judith Rousseau, and Kerrie Mengersen1 posted on arXiv a paper on overfitting mixture models to estimate the number of components. This is directly related with Judith and Kerrie’s 2011 paper and with Zoé’s PhD topic. The paper also returns to the vexing (?) issue of label switching! I very much like the paper and not only because the author are good friends!, but also because it brings a solution to an approach I briefly attempted with Marie-Anne Gruet in the early 1990’s, just before finding about the reversible jump MCMC algorithm of Peter Green at a workshop in Luminy and considering we were not going to “beat the competition”! Hence not publishing the output of our over-fitted Gibbs samplers that were nicely emptying extra components… It also brings a rebuke about a later assertion of mine’s at an ICMS workshop on mixtures, where I defended the notion that over-fitted mixtures could not be detected, a notion that was severely disputed by David McKay…

What is so fantastic in Rousseau and Mengersen (2011) is that a simple constraint on the Dirichlet prior on the mixture weights suffices to guarantee that asymptotically superfluous components will empty out and signal they are truly superfluous! The authors here cumulate the over-fitted mixture with a tempering strategy, which seems somewhat redundant, the number of extra components being a sort of temperature, but eliminates the need for fragile RJMCMC steps. Label switching is obviously even more of an issue with a larger number of components and identifying empty components seems to require a lack of label switching for some components to remain empty!

When reading through the paper, I came upon the condition that only the priors of the weights are allowed to vary between temperatures. Distinguishing the weights from the other parameters does make perfect sense, as some representations of a mixture work without those weights. Still I feel a bit uncertain about the fixed prior constraint, even though I can see the rationale in not allowing for complete freedom in picking those priors. More fundamentally, I am less and less happy with independent identical or exchangeable priors on the components.

Our own recent experience with almost zero weights mixtures (and with Judith, Kaniav, and Kerrie) suggests not using solely a Gibbs sampler there as it shows poor mixing. And even poorer label switching. The current paper does not seem to meet the same difficulties, maybe thanks to (prior) tempering.

The paper proposes a strategy called Zswitch to resolve label switching, which amounts to identify a MAP for each possible number of components and a subsequent relabelling. Even though I do not entirely understand the way the permutation is constructed. I wonder in particular at the cost of the relabelling.