## Correlated Poissons

Posted in Statistics with tags , , on March 2, 2011 by xi'an

A graduate student came to see me the other day with a bivariate Poisson distribution and a question about using EM in this framework. The problem boils down to adding one correlation parameter and an extra term in the likelihood

$(1-\rho)^{n_1}(1+\lambda\rho)^{n_2}(1+\mu\rho)^{n_3}(1-\lambda\mu\rho)^{n_4}\quad 0\le\rho\le\min(1,\frac{1}{\lambda\mu})$

Both terms involving sums are easy to deal with, using latent variables as in mixture models. The subtractions are trickier, as the negative parts cannot appear in a conditional distribution. Even though the problem can be handled by a direct numerical maximisation or by an almost standard Metropolis-within-Gibbs sampler, my suggestion regarding EM per se was to proceed by conditional EM, one parameter at a time. For instance, when considering $\rho$ conditional on both Poisson parameters, depending on whether $\lambda\mu>1$ or not, one can consider either

$(1-\theta/\lambda\mu)^{n_1}(1+\theta/\mu)^{n_2}(1+\theta/\lambda)^{n_3}(1-\theta)^{n_4}\quad0<\theta<1$

and turn

$(1-\theta/\lambda\mu) \text{ into } (1-\theta+\theta\{1-\frac{1}{\lambda\mu}\})$

thus producing a Beta-like target function in $\theta$ after completion, or turn

$(1-\lambda\mu\rho) \text{ into } (1-\rho+\{1-\lambda\mu\}\rho)$

to produce a Beta-like target function in $\rho$ after completion. In the end, this is a rather pedestrian exercise and I am still frustrated at missing the trick to handle the subtractions directly, however it was nonetheless a nice question!

## Computing evidence

Posted in Books, R, Statistics with tags , , , , , , , , , , on November 29, 2010 by xi'an

The book Random effects and latent variable model selection, edited by David Dunson in 2008 as a Springer Lecture Note. contains several chapters dealing with evidence approximation in mixed effect models. (Incidentally, I would be interested in the story behind the  Lecture Note as I found no explanation in the backcover or in the preface. Some chapters but not all refer to a SAMSI workshop on model uncertainty…) The final chapter written by Joyee Ghosh and David Dunson (similar to a corresponding paper in JCGS) contains in particular the interesting identity that the Bayes factor opposing model h to model h-1 can be unbiasedly approximated by (the average of the terms)

$\dfrac{f(x|\theta_{i,h},\mathfrak{M}=h-1)}{f(x|\theta_{i,h},\mathfrak{M}=h)}$

when

• $\mathfrak{M}$ is the model index,
• the $\theta_{i,h}$‘s are simulated from the posterior under model h,
• the model $\mathfrak{M}=h-1$ only considers the h-1 first components of $\theta_{i,h}$,
• the prior under model h-1 is the projection of the prior under model h. (Note that this marginalisation is not the projection used in Bayesian Core.)

## València 9 snapshot [5]

Posted in pictures, Running, Statistics, University life with tags , , , , , , , on June 9, 2010 by xi'an

For the final day of the meeting, after a good one hour run to the end of the Benidorm bay (for me at least!),  we got treated to great talks, culminating with the fitting conclusion given by the conference originator, José Bernardo. The first talk of the day was Guido Consonni’s, who introduced a new class of non-local priors to deal with variable selection. From my understanding, those priors avoid a neighbourhood of zero by placing a polynomial prior on the regression coefficients in order to discriminate better between the null and the alternative,

$\pi(\mathbf{\beta}) = \prod_i \beta_i^ h$

but the influence of the power h seems to be drastic, judging from the example showed by Guido where a move from h=0 to h=1, modified the posterior probability from 0.091 to 0.99 for the same dataset. The discussion by Jim Smith was a perfect finale to the Valencia meetings, Jim being much more abrasive than the usual discussant (while always giving the impression of being near a heart attack//!) The talk from Sylvia Früwirth-Schnatter purposely borrowed Nick Polson’ s title Shrink globally, act locally, and was also dealing with the Bayesian (re)interpretation of Lasso. (I was again left with the impression of hyperparameters that needed to be calibrated but this impression may change after I read the paper!) The talk by Xiao-Li Meng was as efficient as ever with Xiao-Li! Despite the penalising fact of being based on a discussion he wrote for Statistical Science, he managed to convey a global  and convincing picture of likelihood inference in latent variable models, while having the audience laugh most of the talk, a feat repeated by his discussant, Ed George. The basic issue of treating latent variables as parameters offers no particular difficulty in Bayesian inference but this is not true for likelihood models, as shown by both Xiao-Li and Ed. The last talk of the València series managed to make a unifying theory out of the major achievements of José Bernardo and, while I have some criticisms about the outcome, this journey back to decision theory, intrinsic losses and reference priors was nonetheless a very appropriate supplementary contribution of José to this wonderful series of meetings…. Luis Perricchi discussed the paper in a very opinionated manner, defending the role of the Bayes factor, and the debate could have gone forever…Hopefully, I will find time to post my comments on José’s paper.

I am quite sorry I had to leave before the Savage prize session where the four finalists to the prize gave a lecture. Those finalists are of the highest quality as the prize is not given in years when the quality of the theses is not deemed high enough. I will also miss the final evening during which the DeGroot Prize is attributed. (When I received the prize for Bayesian Core. in 2004, I had also left in the morning Valparaiso, just before the banquet!)