Archive for mixtures of distributions

21w5107 [day 2]

Posted in Books, Mountains, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , , on December 1, 2021 by xi'an

After a rich and local (if freezing) dinner on a rooftop facing the baroque Oaxaca cathedral, and an early invigorating outdoor swim in my case!, the morning session was mostly on mixtures, with Helen Ogden exploring X validation for (estimating the number k of components for) finite mixtures, when using the likelihood as an objective function. I was unclear of the goal however when considering that the data supporting the study was Uniform (0,1), nothing like a mixture of Normal distributions. And about the consistency attached to the objective function. The session ended with Diana Cai presenting a counter-argument in the sense that she proved, along with Trevor Campbell and Tamara Broderick, that the posterior on k diverges to infinity with the number n of observations if a mixture model is misspecified for said data. Which does not come as a major surprise since there is no properly defined value of k when the data is not generated from the adopted mixture. I would love to see an extension to the case when the k component mixture contains a non-parametric component! In-between, Alexander Ly discussed Bayes factors for multiple datasets, with some asymptotics showing consistency for some (improper!) priors if one sample size grows to infinity. With actually attaining the same rate under both hypotheses. Luis Nieto-Barajas presented an approach on uncertainty assessment through KL divergence for random probability measures, which requires a calibration of the KL in this setting, as KL does not enjoy a uniform scale, and a prior on a Pólya tree. And Chris Holmes presented a recent work with Edwin Fong and Steven Walker on a prediction approach to Bayesian inference. Which I had had on my reading list for a while. It is a very original proposal where likelihoods and priors are replaced by the sequence of posterior predictives and only parameters of interest get simulated. The Bayesian flavour of the approach is delicate to assess though, albeit a form of non-parametric Bayesian perspective… (I still need to read the paper carefully.)

In the afternoon session, Judith Rousseau presented her recent foray in cut posteriors for semi-parametric HMMs. With interesting outcomes for efficiently estimating the transition matrix, the component distributions, and the smoothing distribution. I wonder at the connection with safe Bayes in that cut posteriors induce a loss of information. Sinead Williamson spoke on distributed MCMC for BNP. Going back at the “theme of the day”, namely clustering and finding the correct (?) number of clusters. With a collapsed versus uncollapsed division that reminded me of the marginal vs. conditional María Gil-Leyva discussed yesterday. Plus a decomposition of a random measure into a finite mixture and an infinite one that also reminded me of the morning talk of Diana Cai. (And making me wonder at the choice of the number K of terms in the finite part.) Michele Guindani spoke about clustering distributions (with firecrackers as a background!). Using the nDP mixture model, which was show to suffer from degeneracy (as discussed by Frederico Camerlenghi et al. in BA). The subtle difference stands in using the same (common) atoms in all random distributions at the top of the hierarchy, with independent weights. Making the partitions partially exchangeable. The approach relies on Sylvia’s generalised mixtures of finite mixtures. With interesting applications to microbiome and calcium imaging (including a mice brain in action!). And Giovanni Rebaudo presented a generalised notion of clustering aligned on a graph, with some observations located between the nodes corresponding to clusters. Represented as a random measure with common parameters for the clusters and separated parameters outside. Interestingly playing on random partitions, Pólya urns, and species sampling.

21w5107 [day 1]

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , , on November 30, 2021 by xi'an

The workshop started by the bad news of our friend Michele Guindani being hit and mugged upon arrival in Oaxaca, Saturday night. Fortunately, he was not hurt, but lost both phone and wallet, always a major bummer when abroad… Still this did not cast a lasting pall on the gathering of long-time no-see friends, whom I had indeed not seen for at least two years. Except for those who came to the CIRMirror!

A few hours later, we got woken up by fairly loud firecrackers (palomas? cohetes?) at 5am, for no reason I can fathom (the Mexican Revolution day was a week ago) although it seemed correlated with the nearby church bells going on at full blast (for Lauds? Hanukkah? Cyber Monday? Chirac’s birthdate?). The above picture was taken the Santa María del Tule town with its super-massive Montezuma cypress tree, with remaining decorations from the Día de los Muertos.

Without launching (much) the debate on whether or not Bayesian non-parametrics qualified as “objective Bayesian” methods, Igor Prünster started the day with a non-parametric presentation of dependent random probability measures. With the always fascinating notion that a random discrete non-parametric prior is inducing a distribution on the partitions (EPPF). And applicability in mixtures and their generalisations. Realising that the highly discrete nature of such measures is not such an issue for a given sample size n, since there are at most n elements in the partition. Beatrice Franzolini discussed of specific ways to create dependent distributions based on independent samples, although her practical example based on one N(-10,1) sample and another (independently) N(10,1) sample seemed to fit in several of the dependent random measures she compared. And Marta Catalano (Warwick) presented her work on partial exchangeability and optimal transportation (which I had also heard in CIRM last June and in Warwick last week). One thing I had not realised earlier was the dependence of the Wasserstein distance on the parameterisation, although it now makes perfect sense. If only for the coupling.  I had alas to miss Isadora Antoniano-Villalobos’ talk as I had to teach my undergrad class in Paris Dauphine at the same time… This non-parametric session was quite homogeneous and rich in perspectives.

In an all-MCMC afternoon, Julyan Arbel talked about reference priors for extreme value distributions, with the “shocking” case of a restriction on the support of one parameter, ξ. Which means in fact that the Jeffreys prior is then undefined. This reminded me somewhat of the work of Clara Grazian on Jeffreys priors for mixtures, where some models were not allowing for Fisher information to exist. The second part of this talk was about modified local versions of Gelman & Rubin (1992) R hats. And the recent modification proposed by Aki and co-authors. Where I thought that a simplification of the multivariate challenge of defining ranks could be alleviated by considering directly the likelihood values of the chains. And Trevor Campbell gradually built an involved parallel tempering method where the powers of a geometric mixture are optimised as spline functions of the temperature. Next, María Gil-Leyva presented her original and ordered approach to mixture estimation, which I discussed in a blog published two days ago (!). She corrected my impressions that (i) the methods were all impervious to label switching and (ii) required some conjugacy to operate. The final talk of the day was by Anirban Bhattacharya on high-D Bayesian regression and coupling techniques for checking convergence, a paper that had been on my reading list for a long while. A very elaborate construct of coupling strategies within a Gibbs sampler, with some steps relying on optimal coupling and others on the use of common random generators.

Handbooks [not a book review]

Posted in Books, pictures, Statistics, University life with tags , , , , , , , on October 26, 2021 by xi'an

mixed feelings

Posted in Books, Kids, Statistics with tags , , , , on September 9, 2021 by xi'an

Two recent questions on X validated about mixtures:

  1. One on the potential negative explosion of the E function in the EM algorithm for a mixture of components with different supports:  “I was hoping to use the EM algorithm to fit a mixture model in which the mixture components can have differing support. I’ve run into a problem during the M step because the expected log-likelihood can be [minus] infinite” Which mistake is based on a confusion between the current parameter estimate and the free parameter to optimise.
  2. Another one on the Gibbs sampler apparently failing for a two-component mixture with only the weights unknown, when the components are close to one another:  “The algorithm works fine if σ is far from 1 but it does not work anymore for σ close to 1.” Which did not see a wide posterior as a possible posterior when both components are similar and hence delicate to distinguish from one another.

 

ISBA 2021 grand finale

Posted in Kids, Mountains, pictures, Running, Statistics, Travel, University life, Wines with tags , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , on July 3, 2021 by xi'an

Last day of ISBA (and ISB@CIRM), or maybe half-day, since there are only five groups of sessions we can attend in Mediterranean time.

My first session was one on priors for mixtures, with 162⁺ attendees at 5:15am! (well, at 11:15 Wien or Marseille time), Gertrud Malsiner-Walli distinguishing between priors on number of components [in the model] vs number of clusters [in the data], with a minor question of mine whether or not a “prior” is appropriate for a data-dependent quantity. And Deborah Dunkel presenting [very early in the US!] anchor models for fighting label switching, which reminded me of the talk she gave at the mixture session of JSM 2018 in Vancouver. (With extensions to consistency and mixtures of regression.) And Clara Grazian debating on objective priors for the number of components in a mixture [in the Sydney evening], using loss functions to build these. Overall it seems there were many talks on mixtures and clustering this year.

After the lunch break, when several ISB@CIRM were about to leave, we ran the Objective Bayes contributed session, which actually included several Stein-like minimaxity talks. Plus one by Théo Moins from the patio of CIRM, with ciccadas in the background. Incredibly chaired by my friend Gonzalo, who had a question at the ready for each and every speaker! And then the Savage Awards II session. Which ceremony is postponed till Montréal next year. And which nominees are uniformly impressive!!! The winner will only be announced in September, via the ISBA Bulletin. Missing the ISBA general assembly for a dinner in Cassis. And being back for the Bayesian optimisation session.

I would have expected more talks at the boundary of BS & ML (as well as COVID and epidemic decision making), the dearth of which should be a cause for concern if researchers at this boundary do not prioritise ISBA meetings over more generic meetings like NeurIPS… (An exception was George Papamakarios’ talk on variational autoencoders in the Savage Awards II session.)

Many many thanks to the group of students at UConn involved in setting most of the Whova site and running the support throughout the conference. It indeed went on very smoothly and provided a worthwhile substitute for the 100% on-site version. Actually, I both hope for the COVID pandemic (or at least the restrictions attached to it) to abate and for the hybrid structure of meetings to stay, along with the multiplication of mirror workshops. Being together is essential to the DNA of conferences, but travelling to a single location is not so desirable, for many reasons. Looking for ISBA 2022, a year from now, either in Montréal, Québec, or in one of the mirror sites!

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