global-local mixtures

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.]

One Response to “global-local mixtures”

  1. Natesh Says:

    Thanks Christian! I think their conjecture 1 is definitely related to our Cauchy result, but not quite the same..

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