Archive for SDEs

séminaire parisien de statistique [09/01/23]

Posted in Books, pictures, Statistics, University life with tags , , , , , , , , , , , , , , , , on January 22, 2023 by xi'an

I had missed the séminaire parisien de statistique for most of the Fall semester, hence was determined to attend the first session of the year 2023, the more because the talks were close to my interest. To wit, Chiara Amorino spoke about particle systems for McKean-Vlasov SDEs, when those are parameterised by several parameters, when observing repeatedly discretised versions, hereby establishing the consistence of a contrast estimator of these estimators. I was initially confused by the mention of interacting particles, since the work is not at all about related with simulation. Just wondering whether this contrast could prove useful for a likelihood-free approach in building a Gibbs distribution?

Valentin de Bortoli then spoke on diffusion Schrödinger bridges for generative models, which allowed me to better my understanding of this idea presented by Arnaud at the Flatiron workshop last November. The presentation here was quite different, using a forward versus backward explanation via a sequence of transforms that end up approximately Gaussian, once more reminiscent of sequential Monte Carlo. The transforms are themselves approximate Gaussian versions relying on adiscretised Ornstein-Ulhenbeck process, with a missing score term since said score involves a marginal density at each step of the sequence. It can be represented [as below] as an expectation conditional on the (observed) variate at time zero (with a connection with Hyvärinen’s NCE / score matching!) Practical implementation is done via neural networks.

Last but not least!, my friend Randal talked about his Kick-Kac formula, which connects with the one we considered in our 2004 paper with Jim Hobert. While I had heard earlier version, this talk was mostly on probability aspects and highly enjoyable as he included some short proofs. The formula is expressing the stationary probability measure π of the original Markov chain in terms of explorations between two visits to an accessible set C, more general than a small set. With at first an annoying remaining term due to the set not being Harris recurrent but which eventually cancels out. Memoryless transportation can be implemented because C is free for the picking, for instance the set where the target is bounded by a manageable density, allowing for an accept-reject step. The resulting chain is non-reversible. However, due to the difficulty to simulate from the target restricted to C, a second and parallel Markov chain is instead created. Performances, unsurprisingly, depend on the choice of C, but it can be adapted to the target on the go.

MCqMC 2014 [day #1]

Posted in pictures, Running, Statistics, Travel, University life with tags , , , , , , , , , on April 9, 2014 by xi'an


As I have been kindly invited to give a talk at MCqMC 2014, here am I. in Leuven, Belgium, for this conference I have never attended before. (I was also invited for MCqMC 2012 in Sydney The talk topics and the attendees’ “sociology” are quite similar to those of the IMACS meeting in Annecy last summer. Namely, rather little on MCMC, particle filters, and other tools familiar in Bayesian computational statistics, but a lot on diffusions and stochastic differential equations and of course quasi-Monte Carlo methods. I thus find myself at a boundary of the conference range and a wee bit lost by some talks, which even titles make little sense to me.

For instance, I have trouble to connect with multi-level Monte Carlo within my own referential. My understanding of the method is one of a control variate version of tempering, namely of using a sequence of approximations to the true target and using rougher approximations as control variates for the finer approximations. But I cannot find on the Web a statistical application of the method outside of diffusions and SDEs, i.e. outside of continuous time processes… Maybe using a particle filter from one approximation to the next, down in terms of roughness, could help.

“Several years ago, Giles (2008) introduced an intriguing multi-level idea to deal with such biased settings that can dramatically improve the rate of convergence and can even, in some settings, achieve the canonical “square root” convergence rate associated with unbiased Monte Carlo.” Rhee and Glynn, 2012

Those were my thoughts before lunchtime. today (namely April 7, 2014). And then, after lunch, Peter Glynn gave his plenary talk that just answered those questions of mine’s!!! Essentially, he showed that formula Pierre Jacob also used in his Bernoulli factory paper to transform a converging-biased-into-an-unbiased estimator, based on a telescopic series representation and a random truncation… This approach is described in a paper with Chang-han Rhee, arXived a few years ago. The talk also covered more recent work (presumably related with Chang-han Rhee’s thesis) extending the above to Markov chains. As explained to me later by Pierre Jacob [of Statisfaction fame!], a regular chain does not converge fast enough to compensate for the explosive behaviour of the correction factor, which is why Rhee and Glynn used instead a backward chain, linking to the exact or perfect samplers of the 1990’s (which origin can be related to a 1992 paper of Asmussen, Glynn and Thorisson). This was certainly the most riveting talk I attended in the past years in that it brought a direct answer to a question I was starting to investigate. And more. I was also wondering how connected it was with our “exact” representation of the stationary distribution (in an Annals of Probability paper with Jim Hobert).   Since we use a stopping rule based on renewal and a geometric waiting time, a somewhat empirical version of the inverse probability found in Peter’s talk. This talk also led me to re-consider a recent discussion we had in my CREST office with Andrew about using square root(ed) importance weights, since one of Peter’s slides exhibited those square roots as optimal. Paradoxically, Peter started the talk by down-playing it, stating there was a single idea therein and a single important slide, making it a perfect after-lunch talk: I wish I had actually had thrice more time to examine each slide! (In the afternoon session, Éric Moulines also gave a thought-provoking talk on particle islands and double bootstrap, a research project I will comment in more detail the day it gets arXived.)

Latent Gaussian Models im Zürich [day 2]

Posted in pictures, R, Statistics, Travel, University life with tags , , , , , , , on February 7, 2011 by xi'an

The second day at the Latent Gaussian Models workshop in Zürich was equally interesting. Among the morning talks, let me mention Daniel Bové who gave a talk connected with the hyper-g prior paper he wrote with Leo Held (commented in an earlier post) and the duo of Janine Illian and Daniel Simpson who gave enthusiastic arguments as to why point pattern datasets should be analysed in a completely novel way, using partial SDEs. And showed us how this could be done via INLA. This perspective (purposedly?) contrasted with the modelling assumptions of Alan Gelfand who concluded the meeting with a highly interesting modelling/estimation of species distribution in the Cape area. He also ran a comparison with the Maxent approach to the same problem. As for my own talk, I somehow spent too much time on the introduction to ABC, trying to link the method with non-parametric perspectives, and so ended rushing through the sufficiency part and the population genetic results obtained by Jean-Marie Cornuet and Jean-Michel Marin the previous day. (The updated slides are available on slideshare.) I hope the main message was still spelled out clearly enough… In conclusion, this was a very interesting workshop, maybe the first of a kind since there is a possible follow-up next year in Trondheim. It showed the clear emergence of a very active INLA community, able to tackle old and new problems using this new technology, and illustrated once again the importance of developing user-friendly codes for promoting such technologies.

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