I was much saddened to hear yesterday that our friend and fellow Bayesian Hélène Massam passed away on August 22, 2020, following a cerebrovascular accident. She was professor of Statistics at York University, in Toronto, and, as her field of excellence covered [the geometry of] exponential families, Wishart distributions and graphical models, we met many times at both Bayesian and nonBayesian conferences (the first time may have been an IMS in Banff, years before BIRS was created). And always had enjoyable conversations on these occasions (in French since she was born in Marseille and only moved to Canada for her graduate studies in optimisation). Beyond her fundamental contributions to exponential families, especially Wishart distributions under different constraints [including the still opened 2007 LetacMassam conjecture], and graphical models, where she produced conjugate priors for DAGs of all sorts, she served the community in many respects, including in the initial editorial board of Bayesian Analysis. I can also personally testify of her dedication as a referee as she helped with many papers along the years. She was also a wonderful person, with a great sense of humor and a love for hiking and mountains. Her demise is a true loss for the entire community and I can only wish her to keep hiking on new planes and cones in a different dimension. [Last month, Christian Genest (McGill University) and Xin Gao (York University) wrote a moving obituary including a complete biography of Hélène for the Statistical Society of Canada.]
Archive for Wishart distribution
Hélène Massam (19492020)
Posted in Statistics with tags 12w5105, Banff, Banff International Research Station for Mathematical Innovation, Bayesian Analysis, BIRS, Canada, DAG, Ecole Normal Supérieure, exponential families, FontenayauxRoses, France, hyperinverse Wishart distribution, ISBA, Marseiile, noncentral Wishart distribution, obituary, Statistical Society of Canada, University of York, Wishart distribution, York on November 1, 2020 by xi'annormalising constants of GWishart densities
Posted in Books, Statistics with tags birthanddeath process, GWishart distribution, graphs, ratio of integrals, reversible jump MCMC, untractable normalizing constant, Wishart distribution on June 28, 2017 by xi'anAbdolreza Mohammadi, Hélène Massam, and Gérard Letac arXived last week a paper on a new approximation of the ratio of two normalising constants associated with two GWishart densities associated with different graphs G. The GWishart is the generalisation of the Wishart distribution by Alberto Roverato to the case when some entries of the matrix are equal to zero, which locations are associated with the graph G. While enjoying the same shape as the Wishart density, this generalisation does not enjoy a closed form normalising constant. Which leads to an intractable ratio of normalising constants when doing Bayesian model selection across different graphs.
AtayKayis and Massam (2005) expressed the ratio as a ratio of two expectations, and the current paper shows that this leads to an approximation of the ratio of normalising constants for a graph G against the graph G augmented by the edge e, equal to
Γ(½{δ+d}) / 2 √π Γ(½{δ+d+1})
where δ is the degree of freedom of the GWishart and d is the number of minimal paths of length 2 linking the two end points of e. This is remarkably concise and provides a fast approximation. (The proof is quite involved, by comparison.) Which can then be used in reversible jump MCMC. The difficulty is obviously in evaluating the impact of the approximation on the target density, as there is no manageable available alternative to calibrate the approximation. In a simulation example where such an alternative is available, the error is negligible though.
multiplying a Gaussian matrix and a Gaussian vector
Posted in Books with tags Bessel functions, blog, cross validated, Gaussian matrix, Laplace distribution, Wishart distribution on March 2, 2017 by xi'anThis arXived note by PierreAlexandre Mattei was actually inspired by one of my blog entries, itself written from a resolution of a question on X validated. The original result about the Laplace distribution actually dates at least to 1932 and a paper by Wishart and Bartlett!I am not sure the construct has clear statistical implications, but it is nonetheless a good calculus exercise.
The note produces an extension to the multivariate case. Where the Laplace distribution is harder to define, in that multiple constructions are possible. The current paper opts for a definition based on the characteristic function. Which leads to a rather unsavoury density with Bessel functions. It however satisfies the constructive definition of being a multivariate Normal multiplied by a χ variate plus a constant vector multiplied by the same squared χ variate. It can also be derived as the distribution of
Wy+y²μ
when W is a (p,q) matrix with iid Gaussian columns and y is a Gaussian vector with independent components. And μ is a vector of the proper dimension. When μ=0 the marginals remain Laplace.
corrected MCMC samplers for multivariate probit models
Posted in Books, pictures, R, Statistics, University life with tags Bayesian modelling, Data augmentation, identifiability, Journal of Econometrics, MNP package, multivariate probit model, probit model, R, Wishart distribution on May 6, 2015 by xi'an“Moreover, IvD point out an error in Nobile’s derivation which can alter its stationary distribution. Ironically, as we shall see, the algorithms of IvD also contain an error.”
Xiyun Jiao and David A. van Dyk arXived a paper correcting an MCMC sampler and R package MNP for the multivariate probit model, proposed by Imai and van Dyk in 2005. [Hence the abbreviation IvD in the above quote.] Earlier versions of the Gibbs sampler for the multivariate probit model by Rob McCulloch and Peter Rossi in 1994, with a Metropolis update added by Agostino Nobile, and finally an improved version developed by Imai and van Dyk in 2005. As noted in the above quote, Jiao and van Dyk have discovered two mistakes in this latest version, jeopardizing the validity of the output.
The multivariate probit model considered here is a multinomial model where the occurrence of the kth category is represented as the kth component of a (multivariate) normal (correlated) vector being the largest of all components. The latent normal model being nonidentifiable since invariant by either translation or scale, identifying constraints are used in the literature. This means using a covariance matrix of the form Σ/trace(Σ), where Σ is an inverse Wishart random matrix. In their 2005 implementation, relying on marginal data augmentation—which essentially means simulating the nonidentifiable part repeatedly at various steps of the data augmentation algorithm—, Imai and van Dyk missed a translation term and a constraint on the simulated matrices that lead to simulations outside the rightful support, as illustrated from the above graph [snapshot from the arXived paper].
Since the IvD method is used in many subsequent papers, it is quite important that these mistakes are signalled and corrected. [Another snapshot above shows how much both algorithm differ!] Without much thinking about this, I [thus idly] wonder why an identifying prior is not taking the place of a hard identifying constraint, as it should solve the issue more nicely. In that it would create less constraints and more entropy (!) in exploring the augmented space, while theoretically providing a convergent approximation of the identifiable parts. I may (must!) however miss an obvious constraint preventing this implementation.
Estimation of covariance matrices
Posted in Statistics with tags arXiv, covariance matrix, JamesStein estimator, Statistical Science, Stein loss, Wishart distribution on June 21, 2011 by xi'anMathilde Bouriga and Olivier Féron have posted a paper on arXiv centred on the estimation of covariance matrices using inverseWishart priors. They introduce hyperpriors on the hyperparameters in the spirit of Daniels and Kass (JASA, 1999) and derive Bayes estimators as well as MCMC procedures. They then run a simulation comparison between the different priors in terms of frequentist risks, concluding in favour of the shrinkage covariance estimators that shrink all components of the empirical covariance matrix. (This paper is part of Mathilde’s thesis, which I coadvise with JeanMichel Marin.)
More among interesting postings on arXiv, many of them published in Statistical Science:

Variable Selection for Nonparametric Gaussian Process Priors: Models and Computational Strategies by Terrance Savitsky, Marina Vannucci, Naijun Sha

Bayesian Statistical Pragmatism by Andrew Gelman (a discussion of the above)

A flexible observed factor model with separate dynamics for the factor volatilities and their correlation matrix by YuCheng Ku, Peter Bloomfield, Robert Kohn

Test martingales, Bayes factors and pvalues by Glenn Shafer, Alexander Shen, Nikolai Vereshchagin, Vladimir Vovk

PitmanYor Diffusion Trees by David A. Knowles, Zoubin Ghahramani