## efficient MCMC sampling

Posted in Statistics with tags , , , on June 24, 2019 by xi'an

Maxime Vono, Daniel Paulin and Arnaud Doucet recently arXived a paper about a regularisation technique that allows for efficient sampling from a complex posterior which potential function factorises as a large sum of transforms of linear projections of the parameter θ

$U(\theta)=\sum_i U_i(A_i\theta)$

The central idea in the paper [which was new to me] is to introduce auxiliary variates for the different terms in the sum, replacing the projections in the transforms, with an additional regularisation forcing these auxiliary variates to be as close as possible from the corresponding projection

$U(\theta,\mathbf z)=\sum_i U_i(z_i)+\varrho^{-1}||z_i-A_i\theta||^2$

This is only an approximation to the true target but it enjoys the possibility to run a massive Gibbs sampler in quite a reduced dimension. As the variance ρ of the regularisation term goes to zero the marginal posterior on the parameter θ converges to the true posterior. The authors manage to achieve precise convergence rates both in total variation and in Wasserstein distance.

From a practical point of view, only judging from the logistic example, it is hard to fathom how much this approach improves upon other approaches (provided they still apply) as the impact of the value of ρ should be assessed on top of the convergence of the high-dimensional Gibbs sampler. Or is there an annealing version in the pipe-line? While parallelisation is a major argument, it also seems that the Gibbs sampler need a central monitoring for each new simulation of θ. Unless some asynchronous version can be implemented.

## delayed but published!

Posted in Statistics with tags , , , , , , on June 20, 2019 by xi'an

## 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 an blog post about the paper on Statisfaction that is definitely much clearer and pedagogical than the above.]

## likelihood free nested sampling

Posted in Books, Statistics with tags , , , , , , , , , , , on April 26, 2019 by xi'an

A recent paper by Mikelson and Khammash found on bioRxiv considers the (paradoxical?) mixture of nested sampling and intractable likelihood. They however cover only the case when a particle filter or another unbiased estimator of the likelihood function can be found. Unless I am missing something in the paper, this seems a very costly and convoluted approach when pseudo-marginal MCMC is available. Or the rather substantial literature on computational approaches to state-space models. Furthermore simulating under the lower likelihood constraint gets even more intricate than for standard nested sampling as the parameter space is augmented with the likelihood estimator as an extra variable. And this makes a constrained simulation the harder, to the point that the paper need resort to a Dirichlet process Gaussian mixture approximation of the constrained density. It thus sounds quite an intricate approach to the problem. (For one of the realistic examples, the authors mention a 12 hour computation on a 48 core cluster. Producing an approximation of the evidence that is not unarguably stabilised, contrary to the above.) Once again, not being completely up-to-date in sequential Monte Carlo, I may miss a difficulty in analysing such models with other methods, but the proposal seems to be highly demanding with respect to the target.

## EntropyMCMC [R package]

Posted in Statistics with tags , , , , , , , , , , , , on March 26, 2019 by xi'an

My colleague from the Université d’Orléans, Didier Chauveau, has just published on CRAN a new R package called EntropyMCMC, which contains convergence assessment tools for MCMC algorithms, based on non-parametric estimates of the Kullback-Leibler divergence between current distribution and target. (A while ago, quite a while ago!, we actually collaborated with a few others on the Springer-Verlag Lecture Note #135 Discretization and MCMC convergence assessments.) This follows from a series of papers by Didier Chauveau and Pierre Vandekerkhove that started with a nearest neighbour entropy estimate. The evaluation of this entropy is based on N iid (parallel) chains, which involves a parallel implementation. While the missing normalising constant is overwhelmingly unknown, the authors this is not a major issue “since we are mostly interested in the stabilization” of the entropy distance. Or in the comparison of two MCMC algorithms. [Disclaimer: I have not experimented with the package so far, hence cannot vouch for its performances over large dimensions or problematic targets, but would as usual welcome comments and feedback on readers’ experiences.]

## BayesComp 20: call for contributed sessions!

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , , , on March 20, 2019 by xi'an

Just to remind readers of the incoming deadline for BayesComp sessions:

The deadline for providing a title and brief abstract that the session is April 1, 2019. Please provide the names and affiliations of the organizer and the three speakers (the organizer can be one of them). Each session lasts 90 minutes and each talk should be 30 minutes long including Q&A. Contributed sessions can also consist of tutorials on the use of novel software. Decisions will be made by April 15, 2019. Please send your proposals to Christian Robert, co-chair of the scientific committee. We look forward to seeing you at BayesComp 20!

In case you do not feel like organising a whole session by yourself, contact the ISBA section you feel affinity with and suggest it helps building this session together!

## a new rule for adaptive importance sampling

Posted in Books, Statistics with tags , , , , , , , , , on March 5, 2019 by xi'an

Art Owen and Yi Zhou have arXived a short paper on the combination of importance sampling estimators. Which connects somehow with the talk about multiple estimators I gave at ESM last year in Helsinki. And our earlier AMIS combination. The paper however makes two important assumptions to reach optimal weighting, which is inversely proportional to the variance:

1. the estimators are uncorrelated if dependent;
2. the variance of the k-th estimator is of order a (negative) power of k.

The later is puzzling when considering a series of estimators, in that k appears to act as a sample size (as in AMIS), the power is usually unknown but also there is no reason for the power to be the same for all estimators. The authors propose to use ½ as the default, both because this is the standard Monte Carlo rate and because the loss in variance is then minimal, being 12% larger.

As an aside, Art Owen also wrote an invited discussion “the unreasonable effectiveness of Monte Carlo” of ” Probabilistic Integration: A Role in Statistical Computation?” by François-Xavier Briol, Chris  Oates, Mark Girolami (Warwick), Michael Osborne and Deni Sejdinovic, to appear in Statistical Science, discussion that contains a wealth of smart and enlightening remarks. Like the analogy between pseudo-random number generators [which work unreasonably well!] vs true random numbers and Bayesian numerical integration versus non-random functions. Or the role of advanced bootstrapping when assessing the variability of Monte Carlo estimates (citing a paper of his from 1992). Also pointing out at an intriguing MCMC paper by  Michael Lavine and Jim Hodges to appear in The American Statistician.