Archive for sequential Monte Carlo

controlled SMC

Posted in Books, pictures, Statistics, University life with tags , , , , , on December 18, 2017 by xi'an

At the end of [last] August, Jeremy Heng, Adrian Bishop†, George Deligiannidis and Arnaud Doucet arXived a paper on controlled sequential Monte Carlo (SMC). That we read today at the BiPs reading group in Paris-Saclay, when I took these notes. The setting is classical SMC, but with a twist in that the proposals at each time iteration are modified by an importance function. (I was quite surprised to discover that this was completely new in that I was under the false impression that it had been tried ages ago!) This importance sampling setting can be interpreted as a change of measures on both the hidden Markov chain and on its observed version. So that the overall normalising constant remains the same. And then being in an importance sampling setting there exists an optimal choice for the importance functions. That results in a zero variance estimated normalising constant, unsurprisingly. And the optimal solution is actually the backward filter familiar to SMC users.

A large part of the paper actually concentrates on figuring out an implementable version of this optimal solution. Using dynamic programming. And projection of each local generator over a simple linear space with Gaussian kernels (aka Gaussian mixtures). Which becomes feasible through the particle systems generated at earlier iterations of said dynamic programming.

The paper is massive, both in terms of theoretical results and of the range of simulations, and we could not get through it within the 90 minutes Sylvain LeCorff spent on presenting it. I can only wonder at this stage how much Rao-Blackwellisation or AMIS could improve the performances of the algorithm. (A point I find quite amazing in Proposition 1 is that the normalising constant Z of the filtering distribution does not change along observations when using the optimal importance function, which translates into the estimates being nearly constant after a few iterations.)

delayed acceptance ABC-SMC

Posted in pictures, Statistics, Travel with tags , , , , , , , on December 11, 2017 by xi'an

Last summer, during my vacation on Skye,  Richard Everitt and Paulina Rowińska arXived a paper on delayed acceptance associated with ABC. ArXival that I missed, then! In order to decrease the number of simulations from the likelihood. As in our own delayed acceptance paper (without ABC), a cheap alternative generator is used to first reject the least likely parameters values, before possibly continuing to use a full generator. Also as lazy ABC. The first step of this ABC algorithm requires a cheap generator plus a primary tolerance ε¹ to compare the generation with the data or part of it. This may be followed by a second generation with a second tolerance level ε². The paper applies more specifically ABC-SMC as introduced in Sisson, Fan and Tanaka (2007) and reassessed in our subsequent 2009 Biometrika paper with Mark Beaumont, Jean-Marie Cornuet and Jean-Michel Marin. As well as in the ABC-SMC paper by Pierre Del Moral and Arnaud Doucet.

When looking at the version of the algorithm [Algorithm 2] based on two basic acceptance ABC steps, there are two features I find intriguing: (i) the primary step uses a cheap generator to reject early poor values of the parameter, followed by the second step involving a more expensive and exact generator, but I see no impact of the choice of this cheap generator in the acceptance probability; (ii) this is an SMC algorithm with imposed resampling at each iteration but there is no visible step for creating new weights after the resampling step. In the current presentation, it sounds like the weights do not change from the initial step, except for those turning to zero and the renormalisation transforms. Which makes the (unspecified) stratification of little interest if any. I must therefore miss a point in the implementation!

One puzzling sentence in the appendix is that the resampling algorithm used in the SMC step “ensures that every particle that is alive before resampling is represented in the resampled particles”, which reminds me of an argument [possibly a different one] made already in Sisson, Fan and Tanaka (2007) and that we could not validate in our subsequent paper. For resampling to be correct, a form of multinomial sampling must be implemented, even via variance reduction schemes like stratified or systematic sampling.

resampling methods

Posted in Books, pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , on December 6, 2017 by xi'an

A paper that was arXived [and that I missed!] last summer is a work on resampling by Mathieu Gerber, Nicolas Chopin (CREST), and Nick Whiteley. Resampling is used to sample from a weighted empirical distribution and to correct for very small weights in a weighted sample that otherwise lead to degeneracy in sequential Monte Carlo (SMC). Since this step is based on random draws, it induces noise (while improving the estimation of the target), reducing this noise is preferable, hence the appeal of replacing plain multinomial sampling with more advanced schemes. The initial motivation is for sequential Monte Carlo where resampling is rife and seemingly compulsory, but this also applies to importance sampling when considering several schemes at once. I remember discussing alternative schemes with Nicolas, then completing his PhD, as well as Olivier Cappé, Randal Douc, and Eric Moulines at the time (circa 2004) we were working on the Hidden Markov book. And getting then a somewhat vague idea as to why systematic resampling failed to converge.

In this paper, Mathieu, Nicolas and Nick show that stratified sampling (where a uniform is generated on every interval of length 1/n) enjoys some form of consistent, while systematic sampling (where the “same” uniform is generated on every interval of length 1/n) does not necessarily enjoy this consistency. There actually exists cases where convergence does not occur. However, a residual version of systematic sampling (where systematic sampling is applied to the residuals of the decimal parts of the n-enlarged weights) is itself consistent.

The paper also studies the surprising feature uncovered by Kitagawa (1996) that stratified sampling applied to an ordered sample brings an error of O(1/n²) between the cdf rather than the usual O(1/n). It took me a while to even understand the distinction between the original and the ordered version (maybe because Nicolas used the empirical cdf during his SAD (Stochastic Algorithm Day!) talk, ecdf that is the same for ordered and initial samples).  And both systematic and deterministic sampling become consistent in this case. The result was shown in dimension one by Kitagawa (1996) but extends to larger dimensions via the magical trick of the Hilbert curve.

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!

the Hyvärinen score is back

Posted in pictures, Statistics, Travel with tags , , , , , , , , , , , , , on November 21, 2017 by xi'an

Stéphane Shao, Pierre Jacob and co-authors from Harvard have just posted on arXiv a new paper on Bayesian model comparison using the Hyvärinen score

\mathcal{H}(y, p) = 2\Delta_y \log p(y) + ||\nabla_y \log p(y)||^2

which thus uses the Laplacian as a natural and normalisation-free penalisation for the score test. (Score that I first met in Padova, a few weeks before moving from X to IX.) Which brings a decision-theoretic alternative to the Bayes factor and which delivers a coherent answer when using improper priors. Thus a very appealing proposal in my (biased) opinion! The paper is mostly computational in that it proposes SMC and SMC² solutions to handle the estimation of the Hyvärinen score for models with tractable likelihoods and tractable completed likelihoods, respectively. (Reminding me that Pierre worked on SMC² algorithms quite early during his Ph.D. thesis.)

A most interesting remark in the paper is to recall that the Hyvärinen score associated with a generic model on a series must be the prequential (predictive) version

\mathcal{H}_T (M) = \sum_{t=1}^T \mathcal{H}(y_t; p_M(dy_t|y_{1:(t-1)}))

rather than the version on the joint marginal density of the whole series. (Followed by a remark within the remark that the logarithm scoring rule does not make for this distinction. And I had to write down the cascading representation

\log p(y_{1:T})=\sum_{t=1}^T \log p(y_t|y_{1:t-1})

to convince myself that this unnatural decomposition, where the posterior on θ varies on each terms, is true!) For consistency reasons.

This prequential decomposition is however a plus in terms of computation when resorting to sequential Monte Carlo. Since each time step produces an evaluation of the associated marginal. In the case of state space models, another decomposition of the authors, based on measurement densities and partial conditional expectations of the latent states allows for another (SMC²) approximation. The paper also establishes that for non-nested models, the Hyvärinen score as a model selection tool asymptotically selects the closest model to the data generating process. For the divergence induced by the score. Even for state-space models, under some technical assumptions.  From this asymptotic perspective, the paper exhibits an example where the Bayes factor and the Hyvärinen factor disagree, even asymptotically in the number of observations, about which mis-specified model to select. And last but not least the authors propose and assess a discrete alternative relying on finite differences instead of derivatives. Which remains a proper scoring rule.

I am quite excited by this work (call me biased!) and I hope it can induce following works as a viable alternative to Bayes factors, if only for being more robust to the [unspecified] impact of the prior tails. As in the above picture where some realisations of the SMC² output and of the sequential decision process see the wrong model being almost acceptable for quite a long while…

postdocs positions in Uppsala in computational stats for machine learning

Posted in Kids, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , on October 22, 2017 by xi'an

Lawrence Murray sent me a call for two postdoc positions in computational statistics and machine learning. In Uppsala, Sweden. With deadline November 17. Definitely attractive for a fresh PhD! Here are some of the contemplated themes:

(1) Developing efficient Bayesian inference algorithms for large-scale latent variable models in data rich scenarios.

(2) Finding ways of systematically combining different inference techniques, such as variational inference, sequential Monte Carlo, and deep inference networks, resulting in new methodology that can reap the benefits of these different approaches.

(3) Developing efficient black-box inference algorithms specifically targeted at inference in probabilistic programs. This line of research may include implementation of the new methods in the probabilistic programming language Birch, currently under development at the department.

Sequential Monte Carlo workshop in Uppsala

Posted in pictures, Statistics, Travel, University life with tags , , , , on May 12, 2017 by xi'an

A workshop on sequential Monte Carlo will take place in Uppsala, Sweeden, on August 30 – September 1, 2017. Involving 21 major players in the field. It follows SMC 2015 that took place at CREST and was organised by Nicolas Chopin. Furthermore, this workshop is preceded by a week-long PhD level course. (The above picture serves as background for the announcement and was taken by Lawrence Murray, whose multiple talents include photography.)