## MCMSki 3 [recollections]

Posted in Mountains, pictures, Statistics, University life with tags , , , , , on January 9, 2011 by xi'an

I am now back home after five exciting and exhausting days in Park City, Utah! As reported earlier, the Adap’skiii meeting went on quite well, with high quality talks relating to edge research. I am thus completely committed to organise the next meeting in three years or so, whether or not MCMSki 4 ever takes place. I also found the ski resort where the meeting took place quite interesting, with plenty of mostly empty ski runs and top quality lodging [with the luxury of a fireplace]. (The downside was a type of runs I was not used to in the Alps, but this showed how far I had to improve in my skiing. And another major downside were the grossly overpriced commodities, because down-town Park City was too far to accommodate my jet-lagged schedule. Despite this lack of complete information, I am slightly bemused at Park City making into the top ten places to go in 2011 according to the New York TImes…) While the picture below, taken from my hotel room/flat, was selected as Shot of the Day by The Canyons, the above panorama picture was provided to me by Luke Bornn, who also gave a fairly interesting talk during the Young Investigators session.

## Poster at MCMSki III

Posted in R, Statistics, Travel with tags , , , , , , , , on December 28, 2010 by xi'an

Here is the poster presented at MCMSki III next week by Pierre Jacob about our joint paper on parallelisation:

## Multidimension bridge sampling (CoRe in CiRM [5])

Posted in Books, R, Statistics, University life with tags , , , , , on July 14, 2010 by xi'an

Since Bayes factor approximation is one of my areas of interest, I was intrigued by Xiao-Li Meng’s comments during my poster in Benidorm that I was using the “wrong” bridge sampling estimator when trying to bridge two models of different dimensions, based on the completion (for $\theta_2=(\mu,\sigma^2)$ and $\mu=\theta_1$ missing from the first model)

$B^\pi_{12}(x)= \dfrac{\displaystyle{\int\pi_1^*(\mu|\sigma^2){\tilde\pi}_1(\sigma^2|x) \alpha(\theta_2) {\pi}_2(\theta_2|x)\hbox{d}\theta_2}}{ \displaystyle{\int{\tilde\pi}_2(\theta_2|x)\alpha(\theta_2) \pi_1(\sigma^2|x)\hbox{d}\sigma^2 } \pi_1^*(\mu|\sigma^2) \hbox{d}\mu }\,.$

When revising the normal chapter of Bayesian Core,  here in CiRM, I thus went back to Xiao-Li’s papers on the topic to try to fathom what the “true” bridge sampling was in that case. In Meng and Schilling (2002, JASA), I found the following indication, “when estimating the ratio of normalizing constants with different dimensions, a good strategy is to bridge each density with a good approximation of itself and then apply bridge sampling to estimate each normalizing constant separately. This is typically more effective than to artificially bridge the two original densities by augmenting the dimension of the lower one”. I was unsure of the technique this (somehow vague) indication pointed at until I understood that it meant  introducing one artificial posterior distribution for each of the parameter spaces and processing each marginal likelihood as an integral ratio in itself. For instance, if $\eta_1(\theta_1)$ is an arbitrary normalised density on $\theta_1$, and $\alpha$ is an arbitrary function, we have the bridge sampling identity on $m_1(x)$:

$\int\tilde{\pi}_1(\theta_1|x) \,\text{d}\theta_1 = \dfrac{\displaystyle{\int \tilde{\pi}_1(\theta_1|x) \alpha(\theta_1) {\eta}_1(\theta_1)\,\text{d}\theta_1}}{\displaystyle{\int\eta_1(\theta_1) \alpha(\theta_1) \pi_1(\theta_1|x) \,\text{d}\theta_1}}$

Therefore, the optimal choice of $\alpha$ leads to the approximation

$\widehat m_1(x) = \dfrac{\displaystyle{\sum_{i=1}^N {\tilde\pi}_1(\theta^\eta_{1i}|x)\big/\left\{{m_1(x) \tilde\pi}_1(\theta^\eta_{1i}|x) + \eta(\theta^\eta_{1i})\right\}}}{\displaystyle{ \sum_{i=1}^{N} \eta(\theta_{1i}) \big/ \left\{{m_1(x) \tilde\pi}_1(\theta_{1i}|x) + \eta(\theta_{1i})\right\}}}$

when $\theta_{1i}\sim\pi_1(\theta_1|x)$ and $\theta^\eta_{1i}\sim\eta(\theta_1)$. More exactly, this approximation is replaced with an iterative version since it depends on the unknown $m_1(x)$. The choice of the density $\eta$ is obviously fundamental and it should be close to the true posterior $\pi_1(\theta_1|x)$ to guarantee good convergence approximation. Using a normal approximation to the posterior distribution of $\theta$ or a non-parametric approximation based on a sample from $\pi_1(\theta_1|\mathbf{x})$, or yet again an average of MCMC proposals are reasonable choices.

The boxplot above compares this solution of Meng and Schilling (2002, JASA), called double (because two pseudo-posteriors $\eta_1(\theta_1)$ and $\eta_2(\theta_2)$ have to be introduced), with Chen, Shao and Ibragim (2001) solution based on a single completion $\pi_1^*$ (using a normal centred at the estimate of the missing parameter, and with variance the estimate from the simulation), when testing whether or not the mean of a normal model with unknown variance is zero. The variabilities are quite comparable in this admittedly overly simple case. Overall, the performances of both extensions are obviously highly dependent on the choice of the completion factors, $\eta_1$ and $\eta_2$ on the one hand and $\pi_1^*$ on the other hand, . The performances of the first solution, which bridges both models via $\pi_1^*$, are bound to deteriorate as the dimension gap between those models increases. The impact of the dimension of the models is less keenly felt for the other solution, as the approximation remains local.

## Ninth València/2010 ISBA World Meeting

Posted in Mountains, Statistics, Travel, University life with tags , , , , , , on May 22, 2010 by xi'an

Just to signal readers that the program of the meeting(s) is now available. It is fairly impressive in its coverage of the ongoing research in Bayesian statistics and related fields, plus it has the very nice feature of completely avoiding parallel sessions, a reason why few contributed talks were accepted. And the less appealing feature of having poster sessions, a highlight of the Valencia meetings, starting at 10pm. Right after the [early Spanish] dinner at 9pm. (As in the earlier meeting in Teneriffe, I will have to find climbing partners for the 1pm-5pm break, even though this is not the best time for climbing…) José Bernardo also indicated that the early registration hotel prices were still in order.

## Savage-Dickey re-rejected

Posted in Statistics, University life with tags , , , , , on January 17, 2010 by xi'an

This morning I found this email in my mailbox,

Dear colleague,

I regret to inform you that your abstract has not been chosen for a talk.  We had 164 submissions for only 36 talks, so the competition was quite high and many good abstracts were not able to be selected. We thank you for your submission, and we hope that you will instead present it as a poster.  If you do want to present a poster, we will need the abstract in the proper format, so please see http://www.bayesian.org/events/isba2010/abstructure.html for format instructions, and then submit your poster abstract to isba10-poster@im.ufrj.br by February 20.

Sincerely,
Herbie Lee, on behalf of the ISBA 10 program committee

which means that our Savage-Dickey paper has been rejected as a topic for a talk at the ISBA 10 conference. Rejected twice in a week, this is not a very promising prospect! (I will obviously resubmit the paper as a poster to ISBA 10, as this is an even better opportunity to argue about our point than a talk. Especially in an environment such as the Valencia meeting where posters occupy a central place in the meeting.)

Post-scriptum: I just registered on the Valencia 9 site and I am rather taken aback by the astounding 485 euro registration fees… Which only cover the fees and the banquet, not the lodging nor the food. There is no reduction for students so it constitutes a big draw on our research funds, maybe justified because this is clearly a major event students should be exposed to, but a big draw nonetheless. Frankly, I cannot explain the amount, except for the  unfortunate choice of the location. If we could have the meeting in a mountainous region like Picos de Europa, we would gain both sides by avoiding the clutter of a mass tourism coastal town and saving on our research money!