Archive for Gaussian mixture

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MCMskv #2 [ridge with a view]

Posted in Mountains, pictures, R, Statistics, Travel, University life with tags , , , , , , , , , , , , , on January 7, 2016 by xi'an

Tuesday at MCMSkv was a rather tense day for me, from having to plan the whole day “away from home” [8km away] to the mundane worry of renting ski equipment and getting to the ski runs over the noon break, to giving a poster over our new mixture paper with Kaniav Kamary and Kate Lee, as Kaniav could not get a visa in time. It actually worked out quite nicely, with almost Swiss efficiency. After Michael Jordan’s talk, I attended a Bayesian molecular biology session with an impressive talk by Jukka Corander on evolutionary genomics with novel ABC aspects. And then a Hamiltonian Monte Carlo session with two deep talks by Sam Livingstone and Elena Akhmatskaya on the convergence of HMC, followed by an amazing entry into Bayesian cosmology by Jens Jasche (with a slight drawback that MCMC simulations took about a calendar year, handling over 10⁷ parameters). Finishing the day with more “classical” MCMC convergence results and techniques, with talks about forgetting time, stopping time (an undervalued alternative to convergence controls), and CLTs. Including a multivariate ESS by James Flegal. (This choice of sessions was uniformly frustrating as I was also equally interested in “the other” session. The drawback of running parallel sessions, obviously.)

The poster session was busy and animated, but I alas could not get an idea of the other posters as I was presenting mine. This was quite exciting as I discussed a new parametrisation for location-scale mixture models that allows for a rather straightforward “non-informative” or reference prior. (The paper with Kaniav Kamary and Kate Lee should be arXived overnight!) The recently deposited CRAN package Ultimixt by Kaniav and Kate contains Metropolis-Hastings functions related to this new approach. The result is quite exciting, especially because I have been looking for it for decades and I will discuss it pretty soon in another post, and I had great exchanges with the conference participants, which led me to consider the reparametrisation in a larger scale and to simplify the presentation of the approach, turning the global mean and variance as hyperparameters.

The day was also most auspicious for a ski break as it was very mild and sunny, while the snow conditions were (somewhat) better than the ones we had in the French Alps two weeks ago. (Too bad that the Tweedie ski race had to be cancelled for lack of snow on the reserved run! The Blossom ski reward will have again to be randomly allocated!) Just not exciting enough to consider another afternoon out, given the tension in getting there and back. (And especially when considering that it took me the entire break time to arXive our mixture paper…)

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no country for odd means

Posted in Books, Kids, Statistics, University life with tags , , , , , , on November 16, 2015 by xi'an

This morning, Clara Grazian and I arXived a paper about Jeffreys priors for mixtures. This is a part of Clara’s PhD dissertation between Roma and Paris, on which she has worked for the past year. Jeffreys priors cannot be computed analytically for mixtures, which is such a drag that it led us to devise the delayed acceptance algorithm. However, the main message from this detailed study of Jeffreys priors is that they mostly do not work for Gaussian mixture models, in that the posterior is almost invariably improper! This is a definite death knell for Jeffreys priors in this setting, meaning that alternative reference priors, like the one we advocated with Kerrie Mengersen and Mike Titterington, or the similar solution in Roeder and Wasserman, have to be used. [Disclaimer: the title has little to do with the paper, except that posterior means are off for mixtures…]

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noninformative priors for mixtures

Posted in Books, Statistics, University life with tags , , , , , , , , on May 26, 2014 by xi'an

“A novel formulation of the mixture model is introduced, which includes the prior constraint that each Gaussian component is always assigned a minimal number of data points. This enables noninformative improper priors such as the Jeffreys prior to be placed on the component parameters. We demonstrate difficulties involved in specifying a prior for the standard Gaussian mixture model, and show how the new model can be used to overcome these. MCMC methods are given for efficient sampling from the posterior of this model.” C. Stoneking

cover of Mixture Estimation and ApplicationsFollowing in the theme of the Jeffreys’ post of two weeks ago, I spotted today a newly arXived paper about using improper priors for mixtures…and surviving it! It is entitled “Bayesian inference of Gaussian mixture models with noninformative priors” and written by Colin Stoneking at ETH Zürich. As mentioned in the previous post, one specificity of our 1990-1994 paper on mixture with Jean Diebolt was to allow for improper priors by imposing at least two observations per component. The above abstract thus puzzled me until I found on page 3 that the paper was indeed related to ours (and Larry’s 2000 validation)! Actually, I should not complain about citations of my earlier works on mixtures as they cover seven different papers, but the bibliography is somewhat missing the paper we wrote with George Casella and Marty Wells in Statistical Methodology in 2004 (this was actually the very first paper of this new journal!), where we show that conjugate priors allow for the integration of the weights, resulting in a close-form expression for the distribution of the partition vector. (This was also extended in the chapter “Exact Bayesian Analysis of Mixtures” I wrote with Kerrie Mengersen in our book Mixtures: Estimation and Applications.)

“There is no well-founded, general method to choose the parameters of a given prior to make it weakly informative for Gaussian mixtures.” C. Stoneking

The first part of the paper shows why looking for weakly informative priors is doomed to fail in this mixture setting: there is no stabilisation as hyperparameters get towards the border (between proper-ness and improper-ness), and on the opposite the frequency of appearances of empty components grows steadily to 100%…  The second part gets to the reassessment of our 1990 exclusion trick, first considering that it is not producing a true posterior, then criticising Larry’s 2000 analysis as building a data-dependent “prior”, and at last proposing a reformulation where the exclusion of the empty components and those with one allocated observation becomes part of the “prior” (albeit a prior on the allocation vector). In fine, the posterior thus constructed remains the same as ours, with a message that if we start our model as the likelihood of the sample excluding empty or single-observation terms, we can produce a proper Bayesian analysis. (Except for a missing if minor renormalisation.) This leads me to wonder about the conclusion that inference about the (unknown) number of components in the mixture being impossible from this perspective. For instance, we could define fractional Bayes factors à la O’Hagan (1995) this way, i.e. starting from the restricted likelihood and taking a fraction of the likelihood to make the posterior proper, then using the remaining fraction to compute a Bayes factor. (Fractional Bayes factors do not work for the regular likelihood of a Gaussian mixture, irrespective of the sample size.)

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a day for comments

Posted in Mountains, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , , , , , , , , , , , on April 21, 2014 by xi'an

As I was flying over Skye (with [maybe] a first if hazy perspective on the Cuillin ridge!) to Iceland, three long sets of replies to some of my posts appeared on the ‘Og:

Thanks to them for taking the time to answer my musings…

 

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MCMC for sampling from mixture models

Posted in Kids, Statistics, University life with tags , , on April 17, 2014 by xi'an

Randal Douc, Florian Maire, and Jimmy Olsson recently arXived a paper on the use of Markov chain Monte Carlo methods for the sampling of mixture models, which contains the recourse to Carlin and Chib (1995) pseudo-priors to simulate from a mixture distribution (and not from the posterior distribution associated with a mixture sampling model). As reported earlier, I was in the thesis defence of Florian Maire and this approach had already puzzled me at the time. In short, a mixture structure

\pi(z)\propto\sum_{m=1}^k \tilde\pi(m,z)

gives rises to as many auxiliary variables as there are components, minus one: namely, if a simulation z is generated from a given component i of the mixture, one can create pseudo-simulations u from all the other components, using pseudo-priors à la Carlin and Chib. A Gibbs sampler based on this augmented state-space can then be implemented:  (a) simulate a new component index m given (z,u);  (b) simulate a new value of (z,u) given m. One version (MCC) of the algorithm simulates z given m from the proper conditional posterior by a Metropolis step, while another one (FCC) only simulate the u‘s. The paper shows that MCC has a smaller asymptotic variance than FCC. I however fail to understand why a Carlin and Chib is necessary in a mixture context: it seems (from the introduction) that the motivation is that a regular Gibbs sampler [simulating z by a Metropolis-Hastings proposal then m] has difficulties moving between components when those components are well-separated. This is correct but slightly moot, as each component of the mixture can be simulated separately and in advance in z, which leads to a natural construction of (a) the pseudo-priors used in the paper, (b) approximations to the weights of the mixture, and (c) a global mixture independent proposal, which can be used in an independent Metropolis-Hastings mixture proposal that [seems to me to] alleviate(s) the need to simulate the component index m. Both examples used in the paper, a toy two-component two-dimensional Gaussian mixture and another toy two-component one-dimensional Gaussian mixture observed with noise (and in absolute value), do not help in perceiving the definitive need for this Carlin and Chib version. Especially when considering the construction of the pseudo-priors.

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