Archive for sequential Monte Carlo

Introduction to Sequential Monte Carlo [book review]

Posted in Books, Statistics with tags , , , , , , , , , , , , , , , , on June 8, 2021 by xi'an

[Warning: Due to many CoI, from Nicolas being a former PhD student of mine, to his being a current colleague at CREST, to Omiros being co-deputy-editor for Biometrika, this review will not be part of my CHANCE book reviews.]

My friends Nicolas Chopin and Omiros Papaspiliopoulos wrote in 2020 An Introduction to Sequential Monte Carlo (Springer) that took several years to achieve and which I find remarkably coherent in its unified presentation. Particles filters and more broadly sequential Monte Carlo have expended considerably in the last 25 years and I find it difficult to keep track of the main advances given the expansive and heterogeneous literature. The book is also quite careful in its mathematical treatment of the concepts and, while the Feynman-Kac formalism is somewhat scary, it provides a careful introduction to the sampling techniques relating to state-space models and to their asymptotic validation. As an introduction it does not go to the same depths as Pierre Del Moral’s 2004 book or our 2005 book (Cappé et al.). But it also proposes a unified treatment of the most recent developments, including SMC² and ABC-SMC. There is even a chapter on sequential quasi-Monte Carlo, naturally connected to Mathieu Gerber’s and Nicolas Chopin’s 2015 Read Paper. Another significant feature is the articulation of the practical part around a massive Python package called particles [what else?!]. While the book is intended as a textbook, and has been used as such at ENSAE and in other places, there are only a few exercises per chapter and they are not necessarily manageable (as Exercise 7.1, the unique exercise for the very short Chapter 7.) The style is highly pedagogical, take for instance Chapter 10 on the various particle filters, with a detailed and separate analysis of the input, algorithm, and output of each of these. Examples are only strategically used when comparing methods or illustrating convergence. While the MCMC chapter (Chapter 15) is surprisingly small, it is actually an introducing of the massive chapter on particle MCMC (and a teaser for an incoming Papaspiloulos, Roberts and Tweedie, a slow-cooking dish that has now been baking for quite a while!).

approximate Bayesian inference [survey]

Posted in Statistics with tags , , , , , , , , , , , , , , , , , , on May 3, 2021 by xi'an

In connection with the special issue of Entropy I mentioned a while ago, Pierre Alquier (formerly of CREST) has written an introduction to the topic of approximate Bayesian inference that is worth advertising (and freely-available as well). Its reference list is particularly relevant. (The deadline for submissions is 21 June,)

sandwiching a marginal

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , on March 8, 2021 by xi'an

When working recently on a paper for estimating the marginal likelihood, I was pointed out this earlier 2015 paper by Roger Grosse, Zoubin Ghahramani and Ryan Adams, which had escaped till now. The beginning of the paper discusses the shortcomings of importance sampling (when simulating from the prior) and harmonic mean (when simulating from the posterior) as solution. And of anNealed importance sampling (when simulating from a sequence, which sequence?!, of targets). The authors are ending up proposing a sequential Monte Carlo or (posterior) particle learning solution. A remark on annealed importance sampling is that there exist both a forward and a backward version for estimating the marginal likelihood, either starting from a simulation from the prior (easy) or from a simulation from the posterior (hard!). As in, e.g., Nicolas Chopin’s thesis, the intermediate steps are constructed from a subsample of the entire sample.

In this context, unbiasedness can be misleading: because partition function estimates can vary over many orders of magnitude, it’s common for an unbiased estimator to drastically underestimate Ζ with overwhelming probability, yet occasionally return extremely large estimates. (An extreme example is likelihood weighting, which is unbiased, but is extremely unlikely to give an accurate answer for a high-dimensional model.) Unless the estimator is chosen very carefully, the variance is likely to be extremely large, or even infinite.”

One novel aspect of the paper is to advocate for the simultaneous use of different methods and for producing both lower and upper bounds on the marginal p(y) and wait for them to get close enough. It is however delicate to find upper bounds, except when using the dreaded harmonic mean estimator.  (A nice trick associated with reverse annealed importance sampling is that the reverse chain can be simulated exactly from the posterior if associated with simulated data, except I am rather lost at the connection between the actual and simulated data.) In a sequential harmonic mean version, the authors also look at the dangers of using an harmonic mean but argue the potential infinite variance of the weights does not matter so much for log p(y), without displaying any variance calculation… The paper also contains a substantial experimental section that compares the different solutions evoked so far, plus others like nested sampling. Which did not work poorly in the experiment (see below) but could not be trusted to provide a lower or an upper bound. The computing time to achieve some level of agreement is however rather daunting. An interesting read definitely (and I wonder what happened to the paper in the end).

the surprisingly overlooked efficiency of SMC

Posted in Books, Statistics, University life with tags , , , , , , , , , , , on December 15, 2020 by xi'an

At the Laplace demon’s seminar today (whose cool name I cannot tire of!), Nicolas Chopin gave a webinar with the above equally cool title. And the first slide debunking myths about SMC’s:

The second part of the talk is about a recent arXival Nicolas wrote with his student Hai-Dang DauI missed, about increasing the number of MCMC steps when moving the particles. Called waste-free SMC. Where only one fraction of the particles is updated, but this is enough to create a sort of independence from previous iterations of the SMC. (Hai-Dang Dau and Nicolas Chopin had to taylor their own convergence proof for this modification of the usual SMC. Producing a single-run assessment of the asymptotic variance.)

On the side, I heard about a very neat (if possibly toyish) example on estimating the number of Latin squares:

And the other item of information is that Nicolas’ and Omiros’ book, An Introduction to Sequential Monte Carlo, has now appeared! (Looking forward reading the parts I had not yet read.)

online approximate Bayesian learning

Posted in Statistics with tags , , , , , , , on September 25, 2020 by xi'an

My 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.