Savage-Dickey revised

We have at last re-submitted (and rearXived, but only to appear on Monday!) our paper on the Savage-Dickey paradox to the Electronic Journal of Statistics (after wasting a few weeks doing nothing!). The revision was quite easy to write, especially because the comments applied to an earlier version of the paper I had submitted by mistake and  theyrequested examples thatactually  were  in the latest version! The comments were in any case quite supportive, although some hinted at a partial misunderstanding with the nature of the “paradox”. I am afraid the measure-theoretic difficulty with this Savage-Dickey paradox will not vanish once the paper is published.

A repulsive random walk

Matt Asher posted an R experiment on R-bloggers yesterday simulating the random walk

which has the property of avoiding zero by quickly switching to a large value as soon as is small. He was then wondering about the “convergence” of the random walk given that it moves very little once is large enough. The values he found for various horizons t seemed to indicate a stable regime.

I reran the same experiment as Matt in a Monte Carlo perspective, using the R program

resu=matrix(0,ncol=100,nrow=25)
sampl=rnorm(100)
for (i in 1:25){
  for (t in 2^(i-1):2^i) sampl=sampl+rnorm(100)/sampl
     resu[i,]=sampl
     }
boxplot(as.data.frame(t(abs(resu))),name=as.character(1:25),col="wheat3")

The outcome of this R code plotted above shows that the range and the average of the 100 replications is increasing with t. This behaviour indicates a transient behaviour of the Markov chain, which almost surely goes to infinity and never comes back (because at infinity the variance is zero). Another indication for transience is shown by the fact that comes back to the interval (-1,1) with probability , a probability which goes to zero with . As suggested to me by Randal Douc, this transience can be established rigorously by considering

which is thus bounded from below by a null recurrent process, which almost surely goes to infinity. Therefore the above Markov chain cannot have a stationary distribution or even a stationary measure: it almost surely goes to (plus or minus) infinity.

Course on adaptive MCMC

Yves Atchadé will give a short course next week on adaptive MCMC methods in École des Ponts (room B413), on June 1, 3, 8 and 9 at 10am. (It sadly clashes with the Valencia meeting for those students who can afford it!) It is free and open to anyone interested. Thus highly recommended.

Significance going X-Atlantic

While reading the latest issue of Significance over breakfast, I discovered that this enjoyable general audience statistics quarterly was about to become jointly distributed by the RSS and the ASA. Given that the magazine is distinctly British, and tuned to the purposes of the Royal Statistical Society, I hope it will remain the way it is. The conditions of the agreement with the ASA are not known to me, so it sounds from the editorial in Significance that it could only be a diffusion agreement. However, the fact that the logo of the ASA will also appear on the cover seems to imply a more active collaboration that could eventually change the tone of the magazine. Let’s wait and see! In the meanwhile, I will enjoy sharing my second copy of Significance with my colleagues (as I could reproduce the comment from a[nother] member of the RSS who said that his spouse, no statistician herself, would take away and read every issue before he could get his hands on it…)

180° south

180° SOUTH is a recent movie repeating Chouinard’s trip to Patagonia about forty years later… (Yvon Chouinard is one of the founders of the mountain brand Patagonia!, who also developped the hexes used in free climbing, later sold by Black Diamond…) Very cool movie, even though I do no see the point in the road trip itself.