**M**y friends and coauthors Matthieu Gerber and Randal Douc have just arXived a massive paper on online approximate Bayesian learning, namely the handling of the posterior distribution on the parameters of a state-space model, which remains a challenge to this day… Starting from the iterated batch importance sampling (IBIS) algorithm of Nicolas (Chopin, 2002) which he introduced in his PhD thesis. The online (“by online we mean that the memory and computational requirement to process each observation is finite and bounded uniformly in t”) method they construct is guaranteed for the approximate posterior to converge to the (pseudo-)true value of the parameter as the sample size grows to infinity, where the sequence of approximations is a Cesaro mixture of initial approximations with Gaussian or t priors, AMIS like. (I am somewhat uncertain about the notion of a sequence of priors used in this setup. Another funny feature is the necessity to consider a fat tail t prior from time to time in this sequence!) The sequence is in turn approximated by a particle filter. The computational cost of this IBIS is roughly in O(NT), depending on the regeneration rate.

## Archive for sequential Monte Carlo

## online approximate Bayesian learning

Posted in Statistics with tags AMIS, approximate MCMC, Bayesian learning, consistency, convergence, ibis, sequential Monte Carlo, state space model on September 25, 2020 by xi'an## nested sampling via SMC

Posted in Books, pictures, Statistics with tags adaptive mixture importance sampling, AMIS, Brisbane, Catch 22, importance sampling, MCMC, multiple importance methods, nested sampling, Queensland, Queensland University of Technology, sequential Monte Carlo, SMC, University of Queensland on April 2, 2020 by xi'an

“We show that by implementing a special type of [sequential Monte Carlo] sampler that takes two im-portance sampling paths at each iteration, one obtains an analogous SMC method to [nested sampling] that resolves its main theoretical and practical issues.”

**A** paper by Queenslander Robert Salomone, Leah South, Chris Drovandi and Dirk Kroese that I had missed (and recovered by Grégoire after we discussed this possibility with our Master students). On using SMC in nested sampling. What are the difficulties mentioned in the above quote?

- Dependence between the simulated samples, since only the offending particle is moved by one or several MCMC steps. (And MultiNest is not a foolproof solution.)
- The error due to quadrature is hard to evaluate, with parallelised versions aggravating the error.
- There is a truncation error due to the stopping rule when the exact maximum of the likelihood function is unknown.

Not mentioning the Monte Carlo error, of course, which should remain at the √n level.

*“N**ested Sampling is a special type of adaptive SMC algorithm, where weights are assigned in a suboptimal way.”*

The above remark is somewhat obvious for a fixed sequence of likelihood levels and a set of particles at each (ring) level. moved by a Markov kernel with the right stationary target. Constrained to move within the ring, which may prove delicate in complex settings. Such a non-adaptive version is however not realistic and hence both the level sets and the stopping rule need be selected from the existing simulation, respectively as a quantile of the observed likelihood and as a failure to modify the evidence approximation, an adaptation that is a Catch 22! as we already found in the AMIS paper. (AMIS stands for adaptive mixture importance sampling.) To escape the quandary, the authors use both an auxiliary variable (to avoid atoms) and two importance sampling sequences (as in AMIS). And only a single particle with non-zero incremental weight for the (upper level) target. As the full details are a bit fuzzy to me, I hope I can experiment with my (quarantined) students on the full implementation of the method.

*“Such cases asides, the question whether SMC is preferable using the TA or NS approach is really one of whether it is preferable to sample (relatively) easy distributions subject to a constraint or to sample potentially difficult distributions.”*

A question (why not regular SMC?) I was indeed considering until coming to the conclusion section but did not find it treated in the paper. There is little discussion on the computing requirements either, as it seems the method is more time-consuming than a regular nested sample. (On the personal side, I appreciated very much their “special thanks to Christian Robert, whose many blog posts on NS helped influence this work, and played a large partin inspiring it.”)

## ensemble rejection sampling

Posted in Statistics with tags accept-reject algorithm, ensemble rejection sampling, importance sampling, sequential Monte Carlo on March 25, 2020 by xi'an**G**eorge Deligiannidis, Arnaud Doucet and Sylvain Rubenthaler have constructed a form of Rao-Blackwellised estimate out of a regular rejection sampler. Doubly surprisingly as turning importance sampling into regular sampling plus gaining over the standard accept-reject estimate. They call their approach ensemble rejection sampling. This is done by seeing the N-sample created from the proposal as an importance sampler, exploiting the importance weights towards estimating the (intractable) normalising constant of the target density, and creating an upper bound on this estimate Ẑ. That depends on the current value X from the N-sample under consideration for acceptance as

Z⁺=Ẑ+{max(w)-w(X)}/N

with a probability Ẑ/Z⁺ to accept X. The amazing result is that the X thus marginaly produced is distributed from the target! Meaning that this is a case for a self-normalised importance sampling distribution producing an exact simulation from the target. While this cannot produce an iid sample, it can be exploited to produce unbiased estimators of expectations under the target. Without even resampling and at a linear cost in the sample size N.

The method can be extended to the dynamic (state-space) case. At a cost of O(N²T) as first observed by Radford Neal. However, the importance sample seems to be distributed from a product of proposals that do not account for the previous particles. But maybe accounting for the observations. While the result involves upper bounds on the dynamic importance weights, the capacity to deliver exact simulations remains a major achievement, in my opinion.

## SMC 2020 [en Madrid]

Posted in pictures, Statistics, Travel, University life with tags CREST, ENSAE, Leganés campus, Madrid, plenary speaker, sequential Monte Carlo, SMC 2020, Spain, uc3m, Universidad Carlos III de Madrid, Uppsala University, workshop on January 30, 2020 by xi'an**A**n announcement for the incoming SMC 2020 workshop, taking place in Madrid next 27-29 of May! The previous workshops were in Paris in 2015 (at ENSAE-CREST) and Uppsala in 2017. This workshop is organised by my friends Víctor Elvira and Joaquín Míguez. With very affordable registration fees and an open call for posters. Here are the invited speakers (so far):

Deniz Akyildiz (University of Warwick)

Christophe Andrieu (University of Bristol)

Nicolas Chopin (ENSAE-CREST)

Dan Crisan (Imperial College London)

Jana de Wiljes (University of Potsdam)

Pierre Del Moral (INRIA)

Petar M. Djuric (Stony Brook University)

Randal Douc (TELECOM SudParis)

Arnaud Doucet (University of Oxford)

Ajay Jasra (National University of Singapore)

Nikolas Kantas (Imperial College London)

Simon Maskell (University of Liverpool)

Lawrence Murray (Uber AI)

François Septier (Université Bretagne Sud)

Sumeetpal Singh (University of Cambridge)

Arno Solin (Aalto University)

Matti Vihola (University of Jyväskylä)

Anna Wigren (Uppsala University)

## postdoc at Warwick on robust SMC [call]

Posted in Kids, pictures, R, Statistics, University life with tags Alan Turing Institute, EPSRC, London, research position, scalable MCMC, sequential Monte Carlo, stochastic algorithms, University of Warwick, Warwick on January 11, 2020 by xi'an**H**ere is a call for a research fellow at the University of Warwick to work with Adam Johansen and Théo Damoulas on the EPSRC and Lloyds Register Foundaton funded project “Robust Scalable Sequential Monte Carlo with application to Urban Air Quality”. To quote

The position will be based primarily at the Department of Statistics of the University of Warwick. The post holder will work closely in collaboration with the rest of the project team and another postdoctoral researcher to be recruited shortly to work within the Data Centric Engineering programme at the Alan Turing Institute in London. The post holder will be expected to visit the Alan Turing Institute regularly.

Candidates with strong backgrounds in the mathematical analysis of stochastic algorithms or sequential Monte Carlo methods are particularly encouraged to apply.Closing date is 19 Jan 2020.