Archive for mixtures of distributions

ratio of Gaussians

Posted in Books, Statistics, University life with tags , , , , , , , , on April 12, 2021 by xi'an

Following (as usual) an X validated question, I came across two papers of George Marsaglia on the ratio of two arbitrary (i.e. unnormalised and possibly correlated) Normal variates. One was a 1965 JASA paper,

where the density of the ratio X/Y is exhibited, based on the fact that this random variable can always be represented as (a+ε)/(b+ξ) where ε,ξ are iid N(0,1) and a,b are constant. Surprisingly (?), this representation was challenged in a 1969 paper by David Hinkley (corrected in 1970).

And less surprisingly the ratio distribution behaves almost like a Cauchy, since its density is

meaning it is a two-component mixture of a Cauchy distribution, with weight exp(-a²/2-b²/2), and of an altogether more complex distribution ƒ². This is remarked by Marsaglia in the second 2006 paper, although the description of the second component remains vague, besides a possible bimodality. (It could have a mean, actually.) The density ƒ² however resembles (at least graphically) the generalised Normal inverse density I played with, eons ago.

handbook of mixture analysis [review]

Posted in Books, R, Statistics with tags , , , , , , , , , on March 19, 2021 by xi'an

“In my opinion, the editors have done an excellent job when selecting the contents of the handbook and putting the different chapters together. For instance, this can be appreciated by the fact that, despite the large number of authors and contributions, all chapters have kept the same notation. Furthermore, in addition to a sound description of the underlying theory and methods, several chapters include information about how to fit the presented models using the R programming language. However, I missed pointers to repositories to download the code and datasets for some of the examples used in the book. To sum up, this is an excellent reference book on mixture models.” Virgilio Gómez-Rubio, JRSS A, 2021

an elegant book [review]

Posted in Books, Statistics, University life with tags , , , , , , , on December 28, 2020 by xi'an

“Handbook of Mixture Analysis is an elegant book on the mixture models. It covers not only statistical foundations but also extensions and applications of mixture models. The book consists of 19 chapters (each chapter is an independent paper), and collectively, these chapters weave into an elegant web of mixture models” Yen-Chi Chen (U. Washington)

stratified MCMC

Posted in Books, pictures, Statistics with tags , , , , , , , , , , , , on December 3, 2020 by xi'an

When working last week with a student, we came across [the slides of a talk at ICERM by Brian van Koten about] a stratified MCMC method whose core idea is to solve a eigenvector equation z’=z’F associated with the masses of “partition” functions Ψ evaluated at the target. (The arXived paper is also available since 2017 but I did not check it in more details.)Although the “partition” functions need to overlap for the matrix not to be diagonal (actually the only case that does not work is when these functions are truly indicator functions). As in other forms of stratified sampling, the practical difficulty is in picking the functions Ψ so that the evaluation of the terms of the matrix F is not overly impacted by the Monte Carlo error. If spending too much time in estimating these terms, there is not a clear gain in switching to stratified sampling, which may be why it is not particularly developed in the MCMC literature….

As an interesting aside, the illustration in this talk comes from the Mexican stamp thickness data I also used in my earlier mixture papers, concerning the 1872 Hidalgo issue that was printed on different qualities of paper. This makes the number k of components somewhat uncertain, although k=3 is sometimes used as a default. Hence a parameter and simulation space of dimension 8, even though the method is used toward approximating the marginal posteriors on the weights λ¹ and λ².


Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , , , on October 15, 2019 by xi'an

There is a conference on mixtures (M) and hidden Markov models (H) and clustering (C) taking place in Orsay on June 17-19, next year. Registration is free if compulsory. With about twenty confirmed speakers. (Irrelevant as the following remark is, this is the opportunity to recall the conference on mixtures I organised in Aussois 25 years before! Which website is amazingly still alive at Duke, thanks to Mike West, my co-organiser along with Kathryn Roeder and Gilles Celeux. When checking the abstracts, I found only two presenters common to both conferences, Christophe Biernaki and Jiahua Chen. And alas several names of departed friends.)