Archive for unbiased MCMC

assessing MCMC convergence

Posted in Books, Statistics, University life with tags , , , , , , , , , , , on June 6, 2019 by xi'an

When MCMC became mainstream in the 1990’s, there was a flurry of proposals to check, assess, and even guarantee convergence to the stationary distribution, as discussed in our MCMC book. Along with Chantal Guihenneuc and Kerrie Mengersen, we also maintained for a while a reviewww webpage categorising theses. Niloy Biswas and Pierre Jacob have recently posted a paper where they propose the use of couplings (and unbiased MCMC) towards deriving bounds on different metrics between the target and the current distribution of the Markov chain. Two chains are created from a given kernel and coupled with a lag of L, meaning that after a while, the two chains become one with a time difference of L. (The supplementary material contains many details on how to induce coupling.) The distance to the target can then be bounded by a sum of distances between the two chains until they merge. The above picture from the paper is a comparison a Polya-Urn sampler with several HMC samplers for a logistic target (not involving the Pima Indian dataset!). The larger the lag L the more accurate the bound. But the larger the lag the more expensive the assessment of how many steps are needed to convergence. Especially when considering that the evaluation requires restarting the chains from scratch and rerunning until they couple again, rather than continuing one run which can only brings the chain closer to stationarity and to being distributed from the target. I thus wonder at the possibility of some Rao-Blackwellisation of the simulations used in this assessment (while realising once more than assessing convergence almost inevitably requires another order of magnitude than convergence itself!). Without a clear idea of how to do it… For instance, keeping the values of the chain(s) at the time of coupling is not directly helpful to create a sample from the target since they are not distributed from that target.

[Pierre also wrote a blog post about the paper on Statisfaction that is definitely much clearer and pedagogical than the above.]

convergences of MCMC and unbiasedness

Posted in pictures, Statistics, University life with tags , , , , , , , , , on January 16, 2018 by xi'an

During his talk on unbiased MCMC in Dauphine today, Pierre Jacob provided a nice illustration of the convergence modes of MCMC algorithms. With the stationary target achieved after 100 Metropolis iterations, while the mean of the target taking much more iterations to be approximated by the empirical average. Plus a nice connection between coupling time and convergence. Convergence to the target.During Pierre’s talk, some simple questions came to mind, from developing an “impatient user version”, as in perfect sampling, in order  to stop chains that run “forever”,  to optimising parallelisation in order to avoid problems of asynchronicity. While the complexity of coupling increases with dimension and the coupling probability goes down, the average coupling time varies but an unexpected figure is that the expected cost per iteration is of 2 simulations, irrespective of the chosen kernels. Pierre also made a connection with optimal transport coupling and stressed that the maximal coupling was for the proposal and not for the target.

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!