## simulating determinantal processes

Posted in Statistics, Travel with tags , , , , , , , , , , on December 6, 2013 by xi'an

In the plane to Atlanta, I happened to read a paper called Efficient simulation of the Ginibre point process by Laurent Decreusefond, Ian Flint, and Anaïs Vergne (from Telecom Paristech). “Happened to” as it was a conjunction of getting tipped by my new Dauphine colleague (and fellow blogger!) Djalil Chaffaï about the paper, having downloaded it prior to departure, and being stuck in a plane (after watching the only Chinese [somewhat] fantasy movie onboard, Saving General Yang).

This is mostly a mathematics paper. While indeed a large chunk of it is concerned with the rigorous definition of this point process in an abstract space, the last part is about simulating such processes. They are called determinantal (and not detrimental as I was tempted to interpret on my first read!) because the density of an n-set (x1x2,…,xn) is given by a kind of generalised Vandermonde determinant

$p(x_1,\ldots,x_n) = \dfrac{1}{n!} \text{det} \left( T(x_i,x_j) \right)$

where T is defined in terms of an orthonormal family,

$T(x,y) = \sum_{i=1}^n \psi_i(x) \overline{\psi_i(y)}.$

(The number n of points can be simulated via an a.s. finite Bernoulli process.) Because of this representation, the sequence of conditional densities for the xi‘s (i.e. x1, x2 given x1, etc.) can be found in closed form. In the special case of the Ginibre process, the ψi‘s are of the form

$\psi_i(z) =z^m \exp\{-|z|^2/2\}/\sqrt{\pi m!}$

and the process cannot be simulated for it has infinite mass, hence an a.s. infinite number of points. Somehow surprisingly (as I thought this was the point of the paper), the authors then switch to a truncated version of the process that always has a fixed number N of points. And whose density has the closed form

$p(x_1,\ldots,x_n) = \dfrac{1}{\pi^N} \prod_i \frac{1}{i!} \exp\{-|z_i|^2/2\}\prod_{i

It has an interestingly repulsive quality in that points cannot get close to one another. (It reminded me of the pinball sampler proposed by Kerrie Mengersen and myself at one of the Valencia meetings and not pursued since.) The conclusion (of this section) is anticlimactic, though,  in that it is known that this density also corresponds to the distribution of the eigenvalues of an Hermitian matrix with standardized complex Gaussian entries. The authors mentions that the fact that the support is the whole complex space Cn is a difficulty, although I do not see why.

The following sections of the paper move to the Ginibre process restricted to a compact and then to the truncated Ginibre process restricted to a compact, for which the authors develop corresponding simulation algorithms. There is however a drag in that the sequence of conditionals, while available in closed-form, cannot be simulated efficiently but rely on a uniform accept-reject instead. While I am certainly missing most of the points in the paper, I wonder if a Gibbs sampler would not be an interesting alternative given that the full (last) conditional is a Gaussian density…

## MCMSki IV (call for proposals)

Posted in Mountains, R, Statistics, University life with tags , , , , on October 15, 2012 by xi'an

The next MCMSki IV conference will for the first time host contributed sessions as well as invited sessions. The scientific committee thus welcomes proposals for contributed talks and even more for contributed sessions. Contributed talks are scheduled to last 20 minutes, plus questions, and contributed sessions one hour and a half, including questions, which corresponds to 4 talks or 3 talks and a discussant.

Proposals should be sent to me, Christian Robert, before March 20, 2013, and includes the name of the speaker(s), the title of the talk(s), and a short 5-15 lines abstract(s). All speakers in a contributed session must be contacted and give their agreement prior to the submission of a session, obviously. The scientific committee will then evaluate the proposals and notify the session organiser/the speaker before April 20, 2013. We remind everyone that MCMSki IV will also schedule two evening poster sessions in the best tradition of the Valencia and MCMSki meetings, sessions in which everyone is welcome to present.

Topics for the proposals include “Big Data issues”, “computationally intensive Bayesian applications”, “probabilistic advances for MC methods”, “variance reduction techniques and Rao-Blackwellisation”, “MC for non-parametric Bayes inference “, “adaptive MC”, “interacting MC”, “INLA”, and “ABC”.

## ISBA 2012, Kyoto [#0]

Posted in pictures, Statistics, Travel, University life with tags , , , , , , on June 25, 2012 by xi'an

The first day of ISBA 2012, Kyoto, consisted of four lectures by Aad van der Vaart, Mike West, Don Berry, and myself. While I had originally understood them to be tutorials in the Valencia tradition, the fact that they were renamed “ISBA lectures on Bayesian foundations” attracted a wide proportion of the conference attendees. I was fairly jet-lagged when giving my talk, but this may have helped to keep the pace at a reasonable level rather than rushing through my 195 slides! There was a party in Gion after the talks in a superb location by the Yasaka Pagoda (unfortunately with Western food!).

This brings me to a call for guest posts: anyone attending ISBA 2012 and wanting to share her or his impressions on the conference, the city, Japanese food or culture, is most welcomed to contact me. Posts will appear here and on the ISBA account of Google+.