## one bridge further

Posted in Books, R, Statistics, University life with tags , , , , , , , , , , , , on June 30, 2020 by xi'an

Jackie Wong, Jon Forster (Warwick) and Peter Smith have just published a paper in Statistics & Computing on bridge sampling bias and improvement by splitting.

“… known to be asymptotically unbiased, bridge sampling technique produces biased estimates in practical usage for small to moderate sample sizes (…) the estimator yields positive bias that worsens with increasing distance between the two distributions. The second type of bias arises when the approximation density is determined from the posterior samples using the method of moments, resulting in a systematic underestimation of the normalizing constant.”

Recall that bridge sampling is based on a double trick with two samples x and y from two (unnormalised) densities f and g that are interverted in a ratio

$m \sum_{i=1}^n g(x_i)\omega(x_i) \Big/ n \sum_{i=1}^m f(y_i)\omega(y_i)$

of unbiased estimators of the inverse normalising constants. Hence biased. The more the less similar these two densities are. Special cases for ω include importance sampling [unbiased] and reciprocal importance sampling. Since the optimal version of the bridge weight ω is the inverse of the mixture of f and g, it makes me wonder at the performance of using both samples top and bottom, since as an aggregated sample, they also come from the mixture, as in Owen & Zhou (2000) multiple importance sampler. However, a quick try with a positive Normal versus an Exponential with rate 2 does not show an improvement in using both samples top and bottom (even when using the perfectly normalised versions)

morc=(sum(f(y)/(nx*dnorm(y)+ny*dexp(y,2)))+
sum(f(x)/(nx*dnorm(x)+ny*dexp(x,2))))/(
sum(g(x)/(nx*dnorm(x)+ny*dexp(x,2)))+
sum(g(y)/(nx*dnorm(y)+ny*dexp(y,2))))


at least in terms of bias… Surprisingly (!) the bias almost vanishes for very different samples sizes either in favour of f or in favour of g. This may be a form of genuine defensive sampling, who knows?! At the very least, this ensures a finite variance for all weights. (The splitting approach introduced in the paper is a natural solution to create independence between the first sample and the second density. This reminded me of our two parallel chains in AMIS.)

## snapshots of Oxford Statistics

Posted in Kids, pictures, Statistics, Travel, University life, Wines with tags , , , , , , , , on February 29, 2016 by xi'an

Following the opening of the new Department of Statistics building in Oxford [which somewhat ironically is the former Department of Mathematics!], a professional photographer was commissioned for a photo cover of this move. Which is incidentally fantastic for the cohesion and work quality of the department, when compared with the former configuration in two disconnected buildings on South Parks Road. Not mentioning the vis-à-vis with Eagle and Child.

As the photographer happened to be there the very day I was teaching my Bayesian module for the OxWaSP PhD students, I ended up in some of the photographs (with no clear memory of this photographer, who was most unintrusive). With my Racoon River Brewing Co. tee-shirt I brought back from Des Moines. And was wearing in a very indirect allusion to the US primaries the night before!

## Rachel’s #1 sunrise in Des Moines

Posted in Kids, pictures, Running, Travel with tags , , , on November 4, 2012 by xi'an

## a day in Chicago

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , , on November 1, 2012 by xi'an

Another busy day as I visited the University of Chicago Booth School of Business. This was my first time visit to this impressive building (and my first visit to Chicago for 25 years…) I actually had to leave Ames at 4:30 (am!) to catch a plane in Des Moines at 6:50 and be at the Chicago Booth before my first appointment at 10… Everything worked out fine, despite the potential for disruption due to the storm Sandy (just spotted a few big waves along the waterfront on my way to the University), and I had a definitely productive sequence of discussions. The talk on ABC was again well-attended and, because this was an econometric seminar (as in Princeton), definitely lively with a flow of questions all along. (There were also a few people from Biology, for whom the focus on our consistency result was presumably less interesting than for econometricians.) As in Ames, I did not manage to reach the part on empirical likelihood. Fodder for another seminar! The day ended by a meal in a superb restaurant with my favourite wine, Saint-Joseph, after which I was ready for a few hours of sleep..! And then a few hours to spend in the Art Institute of Chicago before flying back to Paris. Direct, courtesy of Sandy.

## sunrise in Des Moines

Posted in pictures, Running, Travel with tags , , , on October 29, 2012 by xi'an