## Archive for mixtures of distributions

## van Dantzig seminar

Posted in pictures, Statistics, Travel, University life with tags Amsterdam, Bayes factors, Centrum Wiskunde & Informatica, Chib's approximation, CWI, David van Dantzig, evidence, mixtures of distributions, seminar, Thalys, the Netherlands, Theory of Collective Phenomena, Van Dantzig Seminar on June 3, 2023 by xi'an## Bayesian learning

Posted in Statistics with tags Bayesian learning, Herman Rubin, mixtures of distributions, Newton-Raphson algorithm, One World ABC Seminar, RIKEN, stochastic gradient descent, Tokyo, webinar on May 4, 2023 by xi'an

“…many well-known learning-algorithms, such as those used in optimization, deep learning, and machine learning in general, can now be derived directly following the above scheme using a single algorithm”

**T**he One World ABC webinar today was delivered by Emtiyaz Khan (RIKEN), about the Bayesian Learning Rule, following Khan and Rue 2021 arXival on Bayesian learning. (It had a great intro featuring a video of the speaker’s daughter learning about the purpose of a ukulele in her first year!) The paper argues about a Bayesian interpretation/version of gradient descent algorithms, starting with Zellner’s (1988, the year I first met him!) identity that the posterior is solution to

when ℓ is the likelihood and π the prior. This identity can be generalised to an arbitrary loss function (also dependent on the data) replacing the likelihood and considered for a posterior chosen within an exponential family just as variational Bayes. Ending up with a posterior adapted to this target (in the KL sense). The optimal hyperparameter or pseudo-hyperparameter of this approximation can be recovered by some gradient algorithm, recovering as well stochastic gradient and Newton’s methods. While constructing a prior out of a loss function would have pleased the late Herman Rubin, this is not the case, but rater an approach to deriving a generalised Bayes distribution within a parametric family, including mixtures of Gaussians. At some point in the talk, the uncertainty endemic to the Bayesian approach seeped back into the picture, but since most of the intuition came from machine learning, I was somewhat lost at the nature of this uncertainty.

## efficiency of normalising over discrete parameters

Posted in Statistics with tags arXiv, Gibbs sampler, Hamiltonian Monte Carlo, JAGS, latent variable models, marginalisation, MCMC, mixtures of distributions, Monte Carlo experiment, STAN on May 1, 2022 by xi'an**Y**esterday, I noticed a new arXival entitled *Investigating the efficiency of marginalising over discrete parameters in Bayesian computations* written by Wen Wang and coauthors. The paper is actually comparing the simulation of a Gibbs sampler with an Hamiltonian Monte Carlo approach on Gaussian mixtures, when including and excluding latent variables, respectively. The authors missed the opposite marginalisation when the parameters are integrated.

*While marginalisation requires substantial mathematical effort, folk wisdom in the Stan community suggests that fitting models with marginalisation is more efficient than using Gibbs sampling.*

The comparison is purely experimental, though, which means it depends on the simulated data, the sample size, the prior selection, and of course the chosen algorithms. It also involves the [mostly] automated [off-the-shelf] choices made in the adopted software, JAGS and Stan. The outcome is only evaluated through ESS and the (old) R statistic. Which all depend on the parameterisation. But evacuates the label switching problem by imposing an ordering on the Gaussian means, which may have a different impact on marginalised and unmarginalised models. All in all, there is not much one can conclude about this experiment since the parameter values beyond the simulated data seem to impact the performances much more than the type of algorithm one implements.

## 21w5107 [day 2]

Posted in Books, Mountains, pictures, Statistics, Travel, University life with tags 21w5107, Bayesian nonparametrics, brain imaging, Dirichlet process, Doob's consistency theorem, foundations of objective Bayesian methodology, Jeffreys priors, Kolmogov consistency, MCMC convergence, Mexico, mixtures of distributions, Oaxaca, Oaxaca cuisine, outdoor swimming, parallel MCMC, partial exchangeability, Polya urn, Santa María del Tule, species sampling on December 1, 2021 by xi'an**A**fter 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 21w5107, Día de los Muertos, fireworks, Fisher information, foundations of objective Bayesian methodology, ICML 2021, Jeffreys priors, MCMC convergence, Mexican Revolution day, Mexico, mixtures of distributions, Oaxaca, parallel tempering, random number generator, Santa María del Tule on November 30, 2021 by xi'an**T**he 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.