Archive for MCMC

amazing appendix

Posted in Books, Statistics, Travel, University life with tags , , , , , , , , , , , on February 13, 2018 by xi'an

In the first appendix of the 1995 Statistical Science paper of Besag, Green, Higdon and Mengersen, on MCMC, “Bayesian Computation and Stochastic Systems”, stands a fairly neat result I was not aware of (and which Arnaud Doucet, with his unrivalled knowledge of the literature!, pointed out to me in Oxford, avoiding me the tedium to try to prove it afresco!). I remember well reading a version of the paper in Fort Collins, Colorado, in 1993 (I think!) but nothing about this result.

It goes as follows: when running a Metropolis-within-Gibbs sampler for component x¹ of a collection of variates x¹,x²,…, thus aiming at simulating from the full conditional of x¹ given x⁻¹ by making a proposal q(x|x¹,x⁻¹), it is perfectly acceptable to use a proposal that depends on a parameter α (no surprise so far!) and to generate this parameter α anew at each iteration (still unsurprising as α can be taken as an auxiliary variable) and to have the distribution of this parameter α depending on the other variates x²,…, i.e., x⁻¹. This is the surprising part, as adding α as an auxiliary variable was messing up the update of x⁻¹. But the proof as found in the 1995 paper [page 35] does not require to consider α as such as it establishes global balance directly. (Or maybe still detailed balance when writing the whole Gibbs sampler as a cycle of Metropolis steps.) Terrific! And a whiff mysterious..!

Better together in Kolkata [slides]

Posted in Books, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , on January 4, 2018 by xi'an

Here are the slides of the talk on modularisation I am giving today at the PC Mahalanobis 125 Conference in Kolkata, mostly borrowed from Pierre’s talk at O’Bayes 2018 last month:

[which made me realise Slideshare has discontinued the option to update one’s presentation, forcing users to create a new presentation for each update!] Incidentally, the amphitheatre at ISI is located right on top of a geological exhibit room with a reconstituted Barapasaurus tagorei so I will figuratively ride a dinosaur during my talk!

two Parisian talks by Pierre Jacob in January

Posted in pictures, Statistics, University life with tags , , , , , , , , , on December 21, 2017 by xi'an

While back in Paris from Harvard in early January, Pierre Jacob will give two talks on works of his:

January 09, 10:30, séminaire d’Analyse-Probabilités, Université Paris-Dauphine: Unbiased MCMC

Markov chain Monte Carlo (MCMC) methods provide consistent approximations of integrals as the number of iterations goes to infinity. However, MCMC estimators are generally biased after any fixed number of iterations, which complicates both parallel computation and the construction of confidence intervals. We propose to remove this bias by using couplings of Markov chains and a telescopic sum argument, inspired by Glynn & Rhee (2014). The resulting unbiased estimators can be computed independently in parallel, and confidence intervals can be directly constructed from the Central Limit Theorem for i.i.d. variables. We provide practical couplings for important algorithms such as the Metropolis-Hastings and Gibbs samplers. We establish the theoretical validity of the proposed estimators, and study their variances and computational costs. In numerical experiments, including inference in hierarchical models, bimodal or high-dimensional target distributions, logistic regressions with the Pólya-Gamma Gibbs sampler and the Bayesian Lasso, we demonstrate the wide applicability of the proposed methodology as well as its limitations. Finally, we illustrate how the proposed estimators can approximate the “cut” distribution that arises in Bayesian inference for misspecified models.

January 11, 10:30, CREST-ENSAE, Paris-Saclay: Better together? Statistical learning in models made of modules [Warning: Paris-Saclay is not in Paris!]

In modern applications, statisticians are faced with integrating heterogeneous data modalities relevant for an inference or decision problem. It is convenient to use a graphical model to represent the statistical dependencies, via a set of connected “modules”, each relating to a specific data modality, and drawing on specific domain expertise in their development. In principle, given data, the conventional statistical update then allows for coherent uncertainty quantification and information propagation through and across the modules. However, misspecification of any module can contaminate the update of others. In various settings, particularly when certain modules are trusted more than others, practitioners have preferred to avoid learning with the full model in favor of “cut distributions”. In this talk, I will discuss why these modular approaches might be preferable to the full model in misspecified settings, and propose principled criteria to choose between modular and full-model approaches. The question is intertwined with computational difficulties associated with the cut distribution, and new approaches based on recently proposed unbiased MCMC methods will be described.

Long enough after the New Year festivities (if any) to be fully operational for them!

journée algorithmes stochastiques à Dauphine vendredi

Posted in Statistics, University life with tags , , , , , , , , , , on November 28, 2017 by xi'an

A final reminder (?) that we hold a special day of five talks around stochastic algorithms at Dauphine this Friday, from 10:00 till 17:30. Attendance is free, coffee and tea are free (while they last!), come and join us!

normal variates in Metropolis step

Posted in Books, Kids, R, Statistics, University life with tags , , , , , , , , on November 14, 2017 by xi'an

A definitely puzzled participant on X validated, confusing the Normal variate or variable used in the random walk Metropolis-Hastings step with its Normal density… It took some cumulated efforts to point out the distinction. Especially as the originator of the question had a rather strong a priori about his or her background:

“I take issue with your assumption that advice on the Metropolis Algorithm is useless to me because of my ignorance of variates. I am currently taking an experimental course on Bayesian data inference and I’m enjoying it very much, i believe i have a relatively good understanding of the algorithm, but i was unclear about this specific.”

despite pondering the meaning of the call to rnorm(1)… I will keep this question in store to use in class when I teach Metropolis-Hastings in a couple of weeks.

golden Bayesian!

Posted in Statistics with tags , , , , , , , , , on November 11, 2017 by xi'an

Why is it necessary to sample from the posterior distribution if we already KNOW the posterior distribution?

Posted in Statistics with tags , , , , , , , , on October 27, 2017 by xi'an

I found this question on X validated somewhat hilarious, the more because of the shouted KNOW! And the confused impression that because one can write down π(θ|x) up to a constant, one KNOWS this distribution… It is actually one of the paradoxes of simulation that, from a mathematical perspective, once π(θ|x) is available as a function of (θ,x), all other quantities related with this distribution are mathematically perfectly and uniquely defined. From a numerical perspective, this does not help. Actually, when starting my MCMC course at ENSAE a few days later, I had the same question from a student who thought facing a density function like

f(x) ∞ exp{-||x||²-||x||⁴-||x||⁶}

was enough to immediately produce simulations from this distribution. (I also used this example to show the degeneracy of accept-reject as the dimension d of x increases, using for instance a Gamma proposal on y=||x||. The acceptance probability plunges to zero with d, with 9 acceptances out of 10⁷ for d=20.)