irreversible Markov chains

Werner Krauth (ENS, Paris) was in Dauphine today to present his papers on irreversible Markov chains at the probability seminar. He went back to the 1953 Metropolis et al. paper. And mentioned a 1962 paper I had never heard of by Alder and Wainwright demonstrating phase transition can occur, via simulation. The whole talk was about simulating the stationary distribution of a large number of hard spheres on a one-dimensional ring, which made it hard for me to understand. (Maybe the triathlon before did not help.) And even to realise a part was about PDMPs… His slides included this interesting entry on factorised MCMC which reminded me of delayed acceptance and thinning and prefetching. Plus a notion of lifted Metropolis that could have applications in a general setting, if it differs from delayed rejection.

controlled sequential Monte Carlo [BiPS seminar]

The last BiPS seminar of the semester will be given by Jeremy Heng (Harvard) on Monday 11 June at 2pm, in room 3001, ENSAE, Paris-Saclay about his Controlled sequential Monte Carlo paper:

Sequential Monte Carlo methods, also known as particle methods, are a popular set of techniques to approximate high-dimensional probability distributions and their normalizing constants. They have found numerous applications in statistics and related fields as they can be applied to perform state estimation for non-linear non-Gaussian state space models and Bayesian inference for complex static models. Like many Monte Carlo sampling schemes, they rely on proposal distributions which have a crucial impact on their performance. We introduce here a class of controlled sequential Monte Carlo algorithms, where the proposal distributions are determined by approximating the solution to an associated optimal control problem using an iterative scheme. We provide theoretical analysis of our proposed methodology and demonstrate significant gains over state-of-the-art methods at a fixed computational complexity on a variety of applications.

complex Cauchys

During a visit of Don Fraser and Nancy Reid to Paris-Dauphine where Nancy gave a nice introduction to confidence distributions, Don pointed out to me a 1992 paper by Peter McCullagh on the Cauchy distribution. Following my recent foray into the estimation of the Cauchy location parameter. Among several most interesting aspects of the Cauchy, Peter re-expressed the density of a Cauchy C(θ¹,θ²) as

f(x;θ¹,θ²) = |θ²| / |x-θ|²

when θ=θ¹+ιθ² [a complex number on the half-plane]. Denoting the Cauchy C(θ¹,θ²) as Cauchy C(θ), the property that the ratio aX+b/cX+d follows a Cauchy for all real numbers a,b,c,d,

C(aθ+b/cθ+d)

[when X is C(θ)] follows rather readily. But then comes the remark that

“those properties follow immediately from the definition of the Cauchy as the ratio of two correlated normals with zero mean.”

which seems to relate to the conjecture solved by Natesh Pillai and Xiao-Li Meng a few years ago. But the fact that  a ratio of two correlated centred Normals is Cauchy is actually known at least from the1930’s, as shown by Feller (1930, Biometrika) and Geary (1930, JRSS B).

Bayesian regression trees [seminar]

During her visit to Paris, Veronika Rockovà (Chicago Booth) will give a talk in ENSAE-CREST on the Saclay Plateau at 2pm. Here is the abstract

Posterior Concentration for Bayesian Regression Trees and Ensembles
(joint with Stephanie van der Pas)Since their inception in the 1980’s, regression trees have been one of the more widely used non-parametric prediction methods. Tree-structured methods yield a histogram reconstruction of the regression surface, where the bins correspond to terminal nodes of recursive partitioning. Trees are powerful, yet  susceptible to over-fitting.  Strategies against overfitting have traditionally relied on  pruning  greedily grown trees. The Bayesian framework offers an alternative remedy against overfitting through priors. Roughly speaking, a good prior  charges smaller trees where overfitting does not occur. While the consistency of random histograms, trees and their ensembles  has been studied quite extensively, the theoretical understanding of the Bayesian counterparts has  been  missing. In this paper, we take a step towards understanding why/when do Bayesian trees and their ensembles not overfit. To address this question, we study the speed at which the posterior concentrates around the true smooth regression function. We propose a spike-and-tree variant of the popular Bayesian CART prior and establish new theoretical results showing that  regression trees (and their ensembles) (a) are capable of recovering smooth regression surfaces, achieving optimal rates up to a log factor, (b) can adapt to the unknown level of smoothness and (c) can perform effective dimension reduction when p>n. These results  provide a piece of missing theoretical evidence explaining why Bayesian trees (and additive variants thereof) have worked so well in practice.

distributions for parameters [seminar]

Next Thursday, January 25, Nancy Reid will give a seminar in Paris-Dauphine on distributions for parameters that covers different statistical paradigms and bring a new light on the foundations of statistics. (Coffee is at 10am in the Maths department common room and the talk is at 10:15 in room A, second floor.)

Nancy Reid is University Professor of Statistical Sciences and the Canada Research Chair in Statistical Theory and Applications at the University of Toronto and internationally acclaimed statistician, as well as a 2014 Fellow of the Royal Society of Canada. In 2015, she received the Order of Canada, was elected a foreign associate of the National Academy of Sciences in 2016 and has been awarded many other prestigious statistical and science honours, including the Committee of Presidents of Statistical Societies (COPSS) Award in 1992.

Nancy Reid’s research focuses on finding more accurate and efficient methods to deduce and conclude facts from complex data sets to ultimately help scientists find specific solutions to specific problems.

There is currently some renewed interest in developing distributions for parameters, often without relying on prior probability measures. Several approaches have been proposed and discussed in the literature and in a series of “Bayes, fiducial, and frequentist” workshops and meeting sessions. Confidence distributions, generalized fiducial inference, inferential models, belief functions, are some of the terms associated with these approaches.  I will survey some of this work, with particular emphasis on common elements and calibration properties. I will try to situate the discussion in the context of the current explosion of interest in big data and data science. 

Alan Gelfand in Paris

Alan Gelfand (Duke University) will be in Paris on the week of May 15 and give several seminars, including one at AgroParisTech on May 16:

Modèles hiérarchiques

and on at CREST (BiPS)  on May 18, 2pm:

Scalable Gaussian processes for analyzing space and space-time datasets

talk at Trinity College

Tomorrow noon, I will give a talk at Trinity College Dublin on the asymptotic properties of ABC. (Here are the slides from the talk I gave in Berlin last month.)