Archive for particle filters

Bernoulli race particle filters

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , on March 27, 2019 by xi'an

Sebastian Schmon, Arnaud Doucet and George Deligiannidis have recently arXived an AISTATS paper with the above nice title. The motivation for the extension is facing intractable particle weights for state space models, as for instance in discretised diffusions.  In most cases, actually, the weight associated with the optimal forward proposal involves an intractable integral which is the predictive of the current observed variate given the past hidden states. And in some cases, there exist unbiased and non-negative estimators of the targets,  which can thus be substituted, volens nolens,  to the original filter. As in many pseudo-marginal derivations, this new algorithm can be interpreted as targeting an augmented distribution that involves the auxiliary random variates behind the unbiased estimators of the particle weights. A worthwhile remark since it allows for the preservation of the original target as in (8) provided the auxiliary random variates are simulated from the right conditionals. (At least ideally as I have no clue when this is feasible.)

“if Bernoulli resampling is per-formed, the variance for any Monte Carlo estimate will be the same as if the true weights were known and one applies standard multinomial resampling.”

The Bernoulli race in the title stands for a version of the Bernoulli factory problem, where an intractable and bounded component of the weight can be turned into a probability, for which a Bernoulli draw is available, hence providing a Multinomial sampling with the intractable weights since replacing the exact probability with an estimate does not modify the Bernoulli distribution, amazingly so! Even with intractable normalising constants in particle filters. The practicality of the approach may however be restricted by the possibility of some intractable terms being very small and requiring many rejections for one acceptance, as the number of attempts is a compound geometric. The intractability may add to the time request the drawback of keeping this feature hidden as well. Or force some premature interruption in the settings of a parallel implementation.

IMS workshop [day 5]

Posted in Books, pictures, Statistics, Travel with tags , , , , , , , , on September 3, 2018 by xi'an

The last day of the starting workshop [and my last day in Singapore] was a day of importance [sampling] with talks by Matti Vihola opposing importance sampling and delayed acceptance and particle MCMC, related to several papers of his that I missed. To be continued in the coming weeks at the IMS, which is another reason to regret having to leave that early [as my Parisian semester starts this Monday with an undergrad class at 8:30!]

And then a talk by Joaquín Miguez on stabilizing importance sampling by truncation which reminded me very much of the later work by Andrew Gelman and Aki Vehtari on Pareto smoothed importance sampling, with further operators adapted to sequential settings and the similar drawback that when the importance sampler is poor, i.e., when the simulated points are all very far from the centre of mass, no amount of fudging with the weights will bring the points closer. AMIS made an appearance as a reference method, to be improved by this truncation of the weights, a wee bit surprising as it should bring the large weights of the earlier stages down.

Followed by an almost silent talk by Nick Whiteley, who having lost his voice to the air conditioning whispered his talk in the microphone. Having once faced a lost voice during an introductory lecture to a large undergraduate audience, I could not but completely commiserate for the hardship of the task. Although this made the audience most silent and attentive. His topic was the Viterbi process and its parallelisation, by using a truncated horizon (presenting connection with overdamped Langevin, eg Durmus and Moulines and Dalalyan).

And due to a pressing appointment with my son and his girlfriend [who were traveling through Singapore on that day] for a chili crab dinner on my way to the airport, I missed the final talk by Arnaud Doucet, where he was to reconsider PDMP algorithms without the continuous time layer, a perspective I find most appealing!

Overall, this was a quite diverse and rich [starting] seminar, backed by the superb organisation of the IMS and the smooth living conditions on the NUS campus [once I had mastered the bus routes], which would have made much more sense for me as part of a longer stay, which is actually what happened the previous time I visited the IMS (in 2005), again clashing with my course schedule at home… And as always, I am impressed with the city-state of Singapore, for the highly diverse food scene in particular, but also this [maybe illusory] impression of coexistence between communities. And even though the ecological footprint could certainly be decreased, measures to curb car ownership (with a 150% purchase tax) and use (with congestion charges).

IMS workshop [day 4]

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , , on August 31, 2018 by xi'an

While I did not repeat the mistake of yesterday morning, just as well because the sun was unbearably strong!, I managed this time to board a bus headed in the wrong direction and as a result went through several remote NUS campi! Missing the first talk of the day as a result. By Youssef Marzouk, with a connection between sequential Monte Carlo and optimal transport. Transport for sampling, that is. The following talk by Tiangang Cui was however related, with Marzouk a co-author, as it aimed at finding linear transforms towards creating Normal approximations to the target to be used as proposals in Metropolis algorithms. Which may sound like something already tried a zillion times in the MCMC literature, except that the setting was rather specific to some inverse problems, imposing a generalised Normal structure on the transform, then optimised by transport arguments. It is unclear to me [from just attending the talk] how complex this derivation is and how dimension steps in, but the produced illustrations were quite robust to an increase in dimension.

The remaining talks for the day were mostly particular, from Anthony Lee introducing a new and almost costless way of producing variance estimates in particle filters, exploiting only the ancestry of particles, to Mike Pitt discussing the correlated pseudo-marginal algorithm developed with George Deligiannidis and Arnaud Doucet. Which somewhat paradoxically managed to fight the degeneracy [i.e., the need for a number of terms increasing like the time index T] found in independent pseudo-marginal resolutions, moving down to almost log(T)… With an interesting connection to the quasi SMC approach of Mathieu and Nicolas. And Sebastian Reich also stressed the links with optimal transport in a talk about data assimilation that was way beyond my reach. The day concluded with fireworks, through a magistral lecture by Professeur Del Moral on a continuous time version of PMCMC using the Feynman-Kac terminology. Pierre did a superb job during his lecture towards leading the whole room to the conclusion.

off to Singapore [IMS workshop]

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , on August 26, 2018 by xi'an

Tonight I am off to the National University of Singapore, at the Institute for Mathematical Sciences [and not the Institute of Mathematical Statistics!], to take part in a (first) week workshop on Bayesian Computation for High-Dimensional Statistical Models, covering topics like Approximate Bayesian Computation, Markov chain Monte Carlo, Multilevel Monte Carlo and Particle Filters. Having just barely recovered from the time difference with Vancouver, I now hope I can switch with not too much difficulty to Singapore time zone! As well as face the twenty plus temperature gap with the cool weather this morning in the Parc…

parallelizable sampling method for parameter inference of large biochemical reaction models

Posted in Books, Statistics with tags , , , , , , , , on June 18, 2018 by xi'an

I came across this older (2016) arXiv paper by Jan Mikelson and Mustafa Khammash [antidated as of April 25, 2018] as another version of nested sampling. The novelty of the approach is in applying nested sampling for approximating the likelihood function in the case of involved hidden Markov models (although the name itself does not appear in the paper). This is an interesting proposal, even though there is a fairly large and very active literature on computational approaches to such objects, from sequential Monte Carlo (SMC) to particle MCMC (pMCMC), to SMC².

“We found a way to efficiently sample parameter vectors (particles) from the super level set of the likelihood (sets of particles with a likelihood equal to or higher than some threshold) corresponding to an increasing sequence of thresholds” (p.2)

The approach here is an aggregate of nested sampling and particle filters (SMC), filters that are paradoxically employed in approximating the likelihood function itself, thus called repeatedly as the value of the parameter θ changes, unless I am confused, when it seems to me that, once started with particle filters, the authors could have used them all the way to the upper level (through, again, SMC²). Instead, and that brings a further degree of (uncorrected) approximation to the procedure, a Dirichlet process prior is used to estimate Gaussian mixture approximations to the true posterior distribution(s) on the (super) level sets. Now, approximating a distribution that is zero outside a compact set [the prior restricted to the likelihood being larger than by a distribution with an infinite support does not a priori sound like a particularly enticing idea. Note also that there is no later correction for using the mixture approximation to the restricted prior. (The method also involves an approximation of the (Lebesgue) volume of the level sets that may be poor in higher dimensions.)

“DP-GMM estimations work very well in high dimensional spaces and since we use rejection sampling to obtain samples from the level set by sampling from the DP-GMM estimation, the estimation error does not get propagated through iterations.” (p.13)

One aspect of the paper that puzzles me is the use of a rejection sampler to produce new parameters simulations from a given (super) level set, as this involves a lower bound M on the Gaussian mixture approximation over this level set. If a Gaussian mixture approximation is available, there is apparently no need for this as it can be sampled directly and values below the threshold can be disposed of. It is also unclear why the error does not propagate from one level to the next, if only because of the connection between the successive particle approximations.

 

European statistics in Finland [EMS17]

Posted in Books, pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , , , , , , on August 2, 2017 by xi'an

While this European meeting of statisticians had a wide range of talks and topics, I found it to be more low key than the previous one I attended in Budapest, maybe because there was hardly any talk there in applied probability. (But there were some sessions in mathematical statistics and Mark Girolami gave a great entry to differential geometry and MCMC, in the spirit of his 2010 discussion paper. Using our recent trip to Montréal as an example of geodesic!) In the Bayesian software session [organised by Aki Vetahri], Javier Gonzáles gave a very neat introduction to Bayesian optimisation: he showed how optimisation can be turned into Bayesian inference or more specifically as a Bayesian decision problem using a loss function related to the problem of interest. The point in following a Bayesian path [or probabilist numerics] is to reduce uncertainty by the medium of prior measures on functions, although resorting [as usual] to Gaussian processes whose arbitrariness I somehow dislike within the infinity of priors (aka stochastic processes) on functions! One of his strong arguments was that the approach includes the possibility for design in picking the next observation point (as done in some ABC papers of Michael Guttman and co-authors, incl. the following talk at EMS 2017) but again the devil may be in the implementation when looking at minimising an objective function… The notion of the myopia of optimisation techniques was another good point: only looking one step ahead in the future diminishes the returns of the optimisation and an alternative presented at AISTATS 2016 [that I do not remember seeing in Càdiz] goes against this myopia.

Umberto Piccini also gave a talk on exploiting synthetic likelihoods in a Bayesian fashion (in connection with the talk he gave last year at MCqMC 2016). I wondered at the use of INLA for this Gaussian representation, as well as at the impact of the parameterisation of the summary statistics. And the session organised by Jean-Michel involved Jimmy Olson, Murray Pollock (Warwick) and myself, with great talks from both other speakers, on PaRIS and PaRISian algorithms by Jimmy, and on a wide range of exact simulation methods of continuous time processes by Murray, both managing to convey the intuition behind their results and avoiding the massive mathematics at work there. By comparison, I must have been quite unclear during my talk since someone interrupted me about how Owen & Zhou (2000) justified their deterministic mixture importance sampling representation. And then left when I could not make sense of his questions [or because it was lunchtime already].

estimating constants [survey]

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , on February 2, 2017 by xi'an

A new survey on Bayesian inference with intractable normalising constants was posted on arXiv yesterday by Jaewoo Park and Murali Haran. A rather massive work of 58 pages, almost handy for a short course on the topic! In particular, it goes through the most common MCMC methods with a detailed description, followed by comments on components to be calibrated and the potential theoretical backup. This includes for instance the method of Liang et al. (2016) that I reviewed a few months ago. As well as the Wang-Landau technique we proposed with Yves Atchadé and Nicolas Lartillot. And the noisy MCMC of Alquier et al. (2016), also reviewed a few months ago. (The Russian Roulette solution is only mentioned very briefly as” computationally very expensive”. But still used in some illustrations. The whole area of pseudo-marginal MCMC is also missing from the picture.)

“…auxiliary variable approaches tend to be more efficient than likelihood approximation approaches, though efficiencies vary quite a bit…”

The authors distinguish between MCMC methods where the normalizing constant is approximated and those where it is omitted by an auxiliary representation. The survey also distinguishes between asymptotically exact and asymptotically inexact solutions. For instance, using a finite number of MCMC steps instead of the associated target results in an asymptotically inexact method. The question that remains open is what to do with the output, i.e., whether or not there is a way to correct for this error. In the illustration for the Ising model, the double Metropolis-Hastings version of Liang et al. (2010) achieves for instance massive computational gains, but also exhibits a persistent bias that would go undetected were it the sole method implemented. This aspect of approximate inference is not really explored in the paper, but constitutes a major issue for modern statistics (and machine learning as well, when inference is taken into account.)

In conclusion, this survey provides a serious exploration of recent MCMC methods. It begs for a second part involving particle filters, which have often proven to be faster and more efficient than MCMC methods, at least in state space models. In that regard, Nicolas Chopin and James Ridgway examined further techniques when calling to leave the Pima Indians [dataset] alone.