Archive for transportation model

transport Monte Carlo

Posted in Books, pictures, Statistics, Travel with tags , , , , , , , , , , , , , , , on August 31, 2020 by xi'an

Read this recent arXival by Leo Duan (from UF in Gainesville) on transport approaches to approximate Bayesian computation, in connection with normalising flows. The author points out a “lack of flexibility in a large class of normalizing flows”  to bring forward his own proposal.

“…we assume the reference (a multivariate uniform distribution) can be written as a mixture of many one-to-one transforms from the posterior”

The transportation problem is turned into defining a joint distribution on (β,θ) such that θ is marginally distributed from the posterior and β is one of an infinite collection of transforms of θ. Which sounds quite different from normalizing flows, to be sure. Reverting the order, if one manages to simulate β from its marginal the resulting θ is one of the transforms. Chosen to be a location-scale modification of β, s⊗β+m. The weights when going from θ to β are logistic transforms with Dirichlet distributed scales. All with parameters to be optimised by minimising the Kullback-Leibler distance between the reference measure on β and its inverse mixture approximation, and resorting to gradient descent. (This may sound a wee bit overwhelming as an approximation strategy and I actually had to make a large cup of strong macha to get over it, but this may be due to the heat wave occurring at the same time!) Drawing θ from this approximation is custom-made straightforward and an MCMC correction can even be added, resulting in an independent Metropolis-Hastings version since the acceptance ratio remains computable. Although this may defeat the whole purpose of the exercise by stalling the chain if the approximation is poor (hence suggesting this last step being used instead as a control.)

The paper also contains a theoretical section that studies the approximation error, going to zero as the number of terms in the mixture, K, goes to infinity. Including a Monte Carlo error in log(n)/n (and incidentally quoting a result from my former HoD at Paris 6, Paul Deheuvels). Numerical experiments show domination or equivalence with some other solutions, e.g. being much faster than HMC, the remaining $1000 question being of course the on-line evaluation of the quality of the approximation.

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.

Confronting intractability in Bristol

Posted in pictures, Running, Statistics, Travel, University life, Wines with tags , , , , , , , , , , , , on April 18, 2012 by xi'an

Here are the (revised) slides of my talk this afternoon at the Confronting Intractability in Statistical Inference workshop in Bristol, supported by SuSTain. The novelty is in the final part, where we managed to apply our result to a three population genetic escenario using one versus two δμ summary statistics. This should be the central new example in the incoming revision of our paper to Series B.

More generally, the meeting is very interesting, with great talks and highly relevant topics: e.g., yesterday, I finally understood what transportation models meant (at the general level) and how they related to copula modelling, saw a possible connection from computer models to ABC, got inspiration to mix Gaussian processes with simulation output, and listened to the whole exposition of Simon Wood’s alternative to ABC (much more informative than the four pages of his paper in Nature!). Despite (or due to?) sampling Bath ales yesterday night, I even woke up early enough this morning to run over and under the Clifton suspension bridge, with a slight drizzle that could not really be characterized as rain…