**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!

## Archive for MCMC

## journée algorithmes stochastiques à Dauphine vendredi

Posted in Statistics, University life with tags CEREMADE, computational statistics, MCMC, Paris, sequential Monte Carlo, Statistics, stochastic algorithms, stochastic optimisation, tea, Université Paris Dauphine, workshop on November 28, 2017 by xi'an## normal variates in Metropolis step

Posted in Books, Kids, R, Statistics, University life with tags cross validated, Gaussian random walk, Markov chain Monte Carlo algorithm, MCMC, Metropolis-Hastings algorithm, Monte Carlo Statistical Methods, normal distribution, normal generator, random variates 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 badge, Bayesian, cross validated, introductory textbooks, MCMC, Monte Carlo Statistical Methods, simulation, Stack Exchange, Statistics Forum, wikipedia 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 accept-reject algorithm, course, cross validated, ENSAE, MCMC, Monte Carlo Statistical Methods, posterior distribution, probability, simulation 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.)

## WBIC, practically

Posted in Statistics with tags Bayes factor, Bayesian model selection, evidence, harmonic mean estimator, MCMC, nested sampling, Pima Indians, power posterior, thermodynamic integration, WBIC on October 20, 2017 by xi'an

“Thus far, WBIC has received no more than a cursory mention by Gelman et al. (2013)”

**I** had missed this 2015 paper by Nial Friel and co-authors on a practical investigation of Watanabe’s WBIC. Where WBIC stands for widely applicable Bayesian information criterion. The thermodynamic integration approach explored by Nial and some co-authors for the approximation of the evidence, thermodynamic integration that produces the log-evidence as an integral between temperatures t=0 and t=1 of a powered evidence, is eminently suited for WBIC, as the widely applicable Bayesian information criterion is associated with the specific temperature t⁰ that makes the power posterior equidistant, Kullback-Leibler-wise, from the prior and posterior distributions. And the expectation of the log-likelihood under this very power posterior equal to the (genuine) evidence. In fact, WBIC is often associated with the sub-optimal temperature 1/log(n), where n is the (effective?) sample size. (By comparison, if my minimalist description is unclear!, thermodynamic integration requires a whole range of temperatures and associated MCMC runs.) In an ideal Gaussian setting, WBIC improves considerably over thermodynamic integration, the larger the sample the better. In more realistic settings, though, including a simple regression and a logistic [Pima Indians!] model comparison, thermodynamic integration may do better for a given computational cost although the paper is unclear about these costs. The paper also runs a comparison with harmonic mean and nested sampling approximations. Since the integral of interest involves a power of the likelihood, I wonder if a safe version of the harmonic mean resolution can be derived from simulations of the genuine posterior. Provided the exact temperature t⁰ is known…

## running ABC when the likelihood is available

Posted in Statistics with tags ABC, intractable likelihood, latent variable models, MCMC, MCMC convergence, refereeing on September 19, 2017 by xi'an**T**oday I refereed a paper where the authors used ABC to bypass convergence (and implementation) difficulties with their MCMC algorithm. And I am still pondering whether or not this strategy makes sense. If only because ABC needs to handle the same complexity and the same amount of parameters as an MCMC algorithm. While shooting “in the dark” by using the prior or a coarse substitute to the posterior. And I wonder at the relevance of simulating new data when the [true] likelihood value [at the observed data] can be computed. This would sound to me like the relevant and unique “statistics” worth considering…

## Le Chemin [featuring Randal Douc]

Posted in Books, pictures, Statistics, Travel, University life with tags Cambodia, film, French mathematicians, French movies, MCMC, Randal Douc on September 17, 2017 by xi'an**M**y friend and co-author Randal Douc is one of the main actors in the film Le Chemin that came out last week in French cinemas. Taking place in Cambodia and directed by Jeanne Labrune. I have not yet seen the film but will next week as it is scheduled in a nearby cinema (and only six in Paris!)… (Randal was also a main actor in Rithy Panh’s Un barrage contre le Pacifique, as well as the off-voice in the Oscar nominated Rithy Panh’s L’image manquante.) In connection with this new movie, Randal was interviewed in Allociné, the major French website on current movies. With questions about his future film and theatre projects, but none about his on-going maths research!!!