Filed under: pictures, Travel Tagged: Montpellier, Paris, story, taxi, taxi-driver ]]>

**A post-doctoral position is available in Grenoble, France, to work on computational methods for spatial point process models. The candidate will work with Simon Barthelmé (GIPSA-lab, CNRS) and Jean-François Coeurjolly (Univ. Grenoble Alpes, Laboratory Jean Kuntzmann) on extending point process methodology to deal with large datasets involving multiple sources of variation. We will focus on eye movement data, a new and exciting application area for spatial statistics. The work will take place in the context of an interdisciplinary project on eye movement modelling involving psychologists, statisticians and applied mathematicians from three different institutes in Grenoble.**

The ideal candidate has a background in spatial or computational statistics or machine learning. Knowledge of R (and in particular the package spatstat) and previous experience with point process models is a definite plus.

The duration of the contract is 12+6 months, starting 01.10.2015 at the earliest. Salary is according to standard CNRS scale (roughly EUR 2k/month).

Grenoble is the largest city in the French Alps, with a very strong science and technology cluster. It is a pleasant place to live, in an exceptional mountain environment.

Filed under: Kids, Mountains, Statistics, Travel, University life Tagged: Alps, CNRS, computational statistics, Grenoble, IMAG, Mount Lady Macdonald, mountains, point processes, postdoctoral position, spatial statistics ]]>

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Filed under: pictures, Travel, Wines Tagged: Italia, medieval architecture, San Geminiano, Spring, Tuscany ]]>

In a discrete setting, a way to produce a non-reversible move is to mix the proposal kernel Q with its time-reversed version Q’ and use an acceptance probability of the form

where ε is any weight. This construction is generalised in the paper to any vorticity (skew-symmetric with zero sum rows) matrix Γ, with the acceptance probability

where ε is small enough to ensure all numerator values are non-negative. This is a rather annoying assumption in that, except for the special case derived from the time-reversed kernel, it has to be checked over all pairs (x,y). (I first thought it also implied the normalising constant of π but everything can be set in terms of the unormalised version of π, Γ or ε included.) The paper establishes that the new acceptance probability preserves π as its stationary distribution. An alternative construction is to make the proposal change from Q in H such that H(x,y)=Q(x,y)+εΓ(x,y)/π(x). Which seems more pertinent as not changing the proposal cannot improve that much the mixing behaviour of the chain. Still, the move to the non-reversible versions has the noticeable plus of decreasing the asymptotic variance of the Monte Carlo estimate for any integrable function. Any. (Those results are found in the physics literature of the 2000’s.)

The extension to the continuous case is a wee bit more delicate. One needs to find an anti-symmetric vortex function g with zero integral [equivalent to the row sums being zero] such that g(x,y)+π(y)q(y,x)>0 and with same support as π(x)q(x,y) so that the acceptance probability of g(x,y)+π(y)q(y,x)/π(x)q(x,y) leads to π being the stationary distribution. Once again g(x,y)=ε(π(y)q(y,x)-π(x)q(x,y)) is a natural candidate but it is unclear to me why it should work. As the paper only contains one illustration for the discretised Ornstein-Uhlenbeck model, with the above choice of g for a small enough ε (a point I fail to understand since any ε<1 should provide a positive g(x,y)+π(y)q(y,x)), it is also unclear to me that this modification (i) is widely applicable and (ii) is relevant for genuine MCMC settings.

Filed under: Books, Statistics, University life Tagged: arXiv, MCMC algorithms, Monte Carlo Statistical Methods, Ornstein-Uhlenbeck model, reversibility, Université Paris Dauphine, University of Warwick ]]>

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After the seminar, Christian Lavergne and Jean-Michel had organised a doubly exceptional wine-and-cheese party: first because it is not usually the case there is such a post-seminar party and second because they had chosen a terrific series of wines from the Mas Bruguière (Pic Saint-Loup) vineyards. Ending up with a great 2007 L’Arbouse. Perfect ending for an exciting day. (I am not even mentioning a special Livarot from close to my home-town!)

Filed under: Books, pictures, Statistics, Travel, University life, Wines Tagged: ABC, Bayes factor, Bayesian model choice, Bayesian testing, French cheese, French wines, Languedoc wines, Livarot, Mas Bruguière, Montpellier, Pic Saint Loup ]]>

Filed under: pictures, Travel, Wines Tagged: Castello di Montefioralle, Greve en Chianti, Italia, Italian wines, San Giovese, sunset, Tuscany ]]>

Actually, Natesh and XL further prove an inverse χ² theorem: the inverse of the normal vector, renormalised into a quadratic form is an inverse χ² no matter what its covariance matrix. The proof of this amazing theorem relies on a spherical representation of the bivariate Gaussian (also underlying the Box-Müller algorithm). The angles are then jointly distributed as

and from there follows the argument that conditional on the differences between the θ’s, all ratios are Cauchy distributed. Hence the conclusion!

A question that stems from reading this version of the paper is whether this property extends to other formats of non-independent Cauchy variates. Somewhat connected to my recent post about generating correlated variates from arbitrary distributions: using the inverse cdf transform of a Gaussian copula shows this is possibly the case: the following code is meaningless in that the empirical correlation has no connection with a “true” correlation, but nonetheless the experiment seems of interest…

> ro=.999999;x=matrix(rnorm(2e4),ncol=2);y=ro*x+sqrt(1-ro^2)*matrix(rnorm(2e4),ncol=2) > cor(x[,1]/x[,2],y[,1]/y[,2]) [1] -0.1351967 > ro=.99999999;x=matrix(rnorm(2e4),ncol=2);y=ro*x+sqrt(1-ro^2)*matrix(rnorm(2e4),ncol=2) > cor(x[,1]/x[,2],y[,1]/y[,2]) [1] 0.8622714 > ro=1-1e-5;x=matrix(rnorm(2e4),ncol=2);y=ro*x+sqrt(1-ro^2)*matrix(rnorm(2e4),ncol=2) > z=qcauchy(pnorm(as.vector(x)));w=qcauchy(pnorm(as.vector(y))) > cor(x=z,y=w) [1] 0.9999732 > ks.test((z+w)/2,"pcauchy") One-sample Kolmogorov-Smirnov test data: (z + w)/2 D = 0.0068, p-value = 0.3203 alternative hypothesis: two-sided > ro=1-1e-3;x=matrix(rnorm(2e4),ncol=2);y=ro*x+sqrt(1-ro^2)*matrix(rnorm(2e4),ncol=2) > z=qcauchy(pnorm(as.vector(x)));w=qcauchy(pnorm(as.vector(y))) > cor(x=z,y=w) [1] 0.9920858 > ks.test((z+w)/2,"pcauchy") One-sample Kolmogorov-Smirnov test data: (z + w)/2 D = 0.0036, p-value = 0.9574 alternative hypothesis: two-sided

Filed under: Books, pictures, Running, Statistics, Travel, University life, Wines Tagged: Boston, Box-Muller algorithm, Cauchy distribution, champagne, correlation, Harvard University, JSM 2014, Mathias Drton, Monte Carlo Statistical Methods, Mystic river, Natesh Pillai, Sommerville, Xiao-Li Meng ]]>

Filed under: Kids, pictures, Travel, Wines Tagged: Chianti, Italia, Italian wines, medieval architecture, San Geminiano, Tusca ]]>