Archive for Cauchy

global-local mixtures

Posted in Books, pictures, Running, Statistics, Travel with tags , , on May 4, 2016 by xi'an

Anindya Bhadra, Jyotishka Datta, Nick Polson and Brandon Willard have arXived this morning a short paper on global-local mixtures. Although the definition given in the paper (p.1) is rather unclear, those mixtures are distributions of a sample that are marginals over component-wise (local) and common (global) parameters. The observations of the sample are (marginally) exchangeable if not independent.

“The Cauchy-Schlömilch transformation not only guarantees an ‘astonishingly simple’ normalizing constant for f(·), it also establishes the wide class of unimodal densities as global-local scale mixtures.”

The paper relies on the Cauchy-Schlömilch identity

\int_0^\infty f(\{x-g(x)\}^2)\text{d}x=\int_0^\infty f(y^2)\text{d}y\qquad \text{with}\quad g(x)=g^{-1}(x)

a self-inverse function. This generic result proves helpful in deriving demarginalisations of a Gaussian distribution for densities outside the exponential family like Laplace’s. (This is getting very local for me as Cauchy‘s house is up the hill, while Laplace lived two train stations away. Before train was invented, of course.) And for logistic regression. The paper also briefly mentions Etienne Halphen for his introduction of generalised inverse Gaussian distributions, Halphen who was one of the rare French Bayesians, worked for the State Electricity Company (EDF) and briefly with Lucien Le Cam (before the latter left for the USA). Halphen introduced some families of distributions during the early 1940’s, including the generalised inverse Gaussian family, which were first presented by his friend Daniel Dugué to the Académie des Sciences maybe because of the Vichy racial laws… A second result of interest in the paper is that, given a density g and a transform s on positive real numbers that is decreasing and self-inverse, the function f(x)=2g(x-s(x)) is again a density, which can again be represented as a global-local mixture. [I wonder if these representations could be useful in studying the Cauchy conjecture solved last year by Natesh and Xiao-Li.]

the Poisson transform

Posted in Books, Kids, pictures, Statistics, University life with tags , , , , , , , on June 19, 2014 by xi'an

In obvious connection with an earlier post on the “estimation” of normalising constants, Simon Barthelmé and Nicolas Chopin just arXived a paper on The Poisson transform for unormalised statistical models. Obvious connection because I heard of the Guttmann and Hyvärinen (2012) paper when Simon came to CREST to give a BiP talk on this paper a few weeks ago. (A connected talk he gave in Banff is available as a BIRS video.)

Without getting too much into details, the neat idea therein is to turn the observed likelihood

\sum_{i=1}^n f(x_i|\theta) - n \log \int \exp f(x|\theta) \text{d}x

into a joint likelihood

\sum_{i=1}^n[f(x_i|\theta)+\nu]-n\int\exp[f(x|\theta)+\nu]\text{d}x

 which is the likelihood of a Poisson point process with intensity function

\exp\{ f(x|\theta) + \nu +\log n\}

This is an alternative model in that the original likelihood does not appear as a marginal of the above. Only the modes coincide, with the conditional mode in ν providing the normalising constant. In practice, the above Poisson process likelihood is unavailable and Guttmann and Hyvärinen (2012) offer an approximation by means of their logistic regression.

Unavailable likelihoods inevitably make me think of ABC. Would ABC solutions be of interest there? In particular, could the Poisson point process be simulated with no further approximation? Since the “true” likelihood is not preserved by this representation, similar questions to those found in ABC arise, like a measure of departure from the “true” posterior. Looking forward the Bayesian version! (Marginalia: Siméon Poisson died in Sceaux, which seemed to have attracted many mathematicians at the time, since Cauchy also spent part of his life there…)