Archive for CREST

ABC by QMC

Posted in Books, Kids, Statistics, University life with tags , , , , , , , , , , on November 5, 2018 by xi'an

A paper by Alexander Buchholz (CREST) and Nicolas Chopin (CREST) on quasi-Monte Carlo methods for ABC is going to appear in the Journal of Computational and Graphical Statistics. I had missed the opportunity when it was posted on arXiv and only became aware of the paper’s contents when I reviewed Alexander’s thesis for the doctoral school. The fact that the parameters are simulated (in ABC) from a prior that is quite generally a standard distribution while the pseudo-observations are simulated from a complex distribution (associated with the intractability of the likelihood function) means that the use of quasi-Monte Carlo sequences is in general only possible for the first part.

The ABC context studied there is close to the original version of ABC rejection scheme [as opposed to SMC and importance versions], the main difference standing with the use of M pseudo-observations instead of one (of the same size as the initial data). This repeated version has been discussed and abandoned in a strict Monte Carlo framework in favor of M=1 as it increases the overall variance, but the paper uses this version to show that the multiplication of pseudo-observations in a quasi-Monte Carlo framework does not increase the variance of the estimator. (Since the variance apparently remains constant when taking into account the generation time of the pseudo-data, we can however dispute the interest of this multiplication, except to produce a constant variance estimator, for some targets, or to be used for convergence assessment.) L The article also covers the bias correction solution of Lee and Latuszyǹski (2014).

Due to the simultaneous presence of pseudo-random and quasi-random sequences in the approximations, the authors use the notion of mixed sequences, for which they extend a one-dimension central limit theorem. The paper focus on the estimation of Z(ε), the normalization constant of the ABC density, ie the predictive probability of accepting a simulation which can be estimated at a speed of O(N⁻¹) where N is the number of QMC simulations, is a wee bit puzzling as I cannot figure the relevance of this constant (function of ε), especially since the result does not seem to generalize directly to other ABC estimators.

A second half of the paper considers a sequential version of ABC, as in ABC-SMC and ABC-PMC, where the proposal distribution is there  based on a Normal mixture with a small number of components, estimated from the (particle) sample of the previous iteration. Even though efficient techniques for estimating this mixture are available, this innovative step requires a calculation time that should be taken into account in the comparisons. The construction of a decreasing sequence of tolerances ε seems also pushed beyond and below what a sequential approach like that of Del Moral, Doucet and Jasra (2012) would produce, it seems with the justification to always prefer the lower tolerances. This is not necessarily the case, as recent articles by Li and Fearnhead (2018a, 2018b) and ours have shown (Frazier et al., 2018). Overall, since ABC methods are large consumers of simulation, it is interesting to see how the contribution of QMC sequences results in the reduction of variance and to hope to see appropriate packages added for standard distributions. However, since the most consuming part of the algorithm is due to the simulation of the pseudo-data, in most cases, it would seem that the most relevant focus should be on QMC add-ons on this part, which may be feasible for models with a huge number of standard auxiliary variables as for instance in population evolution.

clustering dynamical networks

Posted in pictures, Statistics, University life with tags , , , , , , , , , , on June 5, 2018 by xi'an


Yesterday I attended a presentation by Catherine Matias on dynamic graph structures, as she was giving a plenary talk at the 50th French statistical meeting, conveniently located a few blocks away from my office at ENSAE-CREST. In the nicely futuristic buildings of the EDF campus, which are supposed to represent cogs according to the architect, but which remind me more of these gas holders so common in the UK, at least in the past! (The E of EDF stands for electricity, but the original public company handled both gas and electricity.) This was primarily a survey of the field, which is much more diverse and multifaceted than I realised, even though I saw some recent developments by Antonietta Mira and her co-authors, as well as refereed a thesis on temporal networks at Ca’Foscari by Matteo Iacopini, which defence I will attend in early July. The difficulty in the approaches covered by Catherine stands with the amount and complexity of the latent variables induced by the models superimposed on the data. In her paper with Christophe Ambroise, she followed a variational EM approach. From the spectator perspective that is mine, I wondered at using ABC instead, which is presumably costly when the data size grows in space or in time. And at using tensor structures as in Mateo’s thesis. This reminded me as well of Luke Bornn’s modelling of basketball games following each player in real time throughout the game. (Which does not prevent the existence of latent variables.) But more vaguely and speculatively I also wonder at the meaning of the chosen models, which try to represent “everything” in the observed process, which seems doomed from the start given the heterogeneity of the data. While reaching my Keynesian pessimistic low- point, which happens rather quickly!, one could hope for projection techniques, towards reducing the dimension of the data of interest and of the parameter required by the model.

rage against the [Nature] Machine [Intelligence]

Posted in Books, Statistics, University life with tags , , , , , , , , , on May 15, 2018 by xi'an

Yesterday evening, my friend and colleague Pierre Alquier (CREST-ENSAE) got interviewed (for a few seconds on-line!, around minute 06) by the French national radio, France Culture, about the recent call to boycott the incoming Nature Machine Intelligence electronic journal. Call to the machine learning community, based on the lack of paying journals among the major machine learnings journals, like JMLR. Meaning that related conferences like AISTATS and NIPS also get their accepted papers available on-line for free. As noted in the call

“Machine learning has been at the forefront of the movement for free and open access to research. For example, in 2001 the Editorial Board of the Machine Learning Journal resigned en masse to form a new zero-cost open access journal, the Journal of Machine Learning Research (JMLR).”

Bayesian regression trees [seminar]

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

two Parisian talks by Pierre Jacob in January

Posted in pictures, Statistics, University life with tags , , , , , , , , , on December 21, 2017 by xi'an

While back in Paris from Harvard in early January, Pierre Jacob will give two talks on works of his:

January 09, 10:30, séminaire d’Analyse-Probabilités, Université Paris-Dauphine: Unbiased MCMC

Markov chain Monte Carlo (MCMC) methods provide consistent approximations of integrals as the number of iterations goes to infinity. However, MCMC estimators are generally biased after any fixed number of iterations, which complicates both parallel computation and the construction of confidence intervals. We propose to remove this bias by using couplings of Markov chains and a telescopic sum argument, inspired by Glynn & Rhee (2014). The resulting unbiased estimators can be computed independently in parallel, and confidence intervals can be directly constructed from the Central Limit Theorem for i.i.d. variables. We provide practical couplings for important algorithms such as the Metropolis-Hastings and Gibbs samplers. We establish the theoretical validity of the proposed estimators, and study their variances and computational costs. In numerical experiments, including inference in hierarchical models, bimodal or high-dimensional target distributions, logistic regressions with the Pólya-Gamma Gibbs sampler and the Bayesian Lasso, we demonstrate the wide applicability of the proposed methodology as well as its limitations. Finally, we illustrate how the proposed estimators can approximate the “cut” distribution that arises in Bayesian inference for misspecified models.

January 11, 10:30, CREST-ENSAE, Paris-Saclay: Better together? Statistical learning in models made of modules [Warning: Paris-Saclay is not in Paris!]

In modern applications, statisticians are faced with integrating heterogeneous data modalities relevant for an inference or decision problem. It is convenient to use a graphical model to represent the statistical dependencies, via a set of connected “modules”, each relating to a specific data modality, and drawing on specific domain expertise in their development. In principle, given data, the conventional statistical update then allows for coherent uncertainty quantification and information propagation through and across the modules. However, misspecification of any module can contaminate the update of others. In various settings, particularly when certain modules are trusted more than others, practitioners have preferred to avoid learning with the full model in favor of “cut distributions”. In this talk, I will discuss why these modular approaches might be preferable to the full model in misspecified settings, and propose principled criteria to choose between modular and full-model approaches. The question is intertwined with computational difficulties associated with the cut distribution, and new approaches based on recently proposed unbiased MCMC methods will be described.

Long enough after the New Year festivities (if any) to be fully operational for them!

Les Rouquins

Posted in pictures, University life, Wines with tags , , , , , , , on October 20, 2017 by xi'an

Langevin on a wrong bend

Posted in Books, Statistics with tags , , , , , , , on October 19, 2017 by xi'an

Arnak Dalayan and Avetik Karagulyan (CREST) arXived a paper the other week on a focussed study of the Langevin algorithm [not MALA] when the gradient of the target is incorrect. With the following improvements [quoting non-verbatim from the paper]:

  1. a varying-step Langevin that reduces the number of iterations for a given Wasserstein precision, compared with recent results by e.g. Alan Durmus and Éric Moulines;
  2. an extension of convergence results for error-prone evaluations of the gradient of the target (i.e., the gradient is replaced with a noisy version, under some moment assumptions that do not include unbiasedness);
  3. a new second-order sampling algorithm termed LMCO’, with improved convergence properties.

What is particularly interesting to me in this setting is the use in all these papers of a discretised Langevin diffusion (a.k.a., random walk with a drift induced by the gradient of the log-target) without the original Metropolis correction. The results rely on an assumption of [strong?] log-concavity of the target, with “user-friendly” bounds on the Wasserstein distance depending on the constants appearing in this log-concavity constraint. And so does the adaptive step. (In the case of the noisy version, the bias and variance of the noise also matter. As pointed out by the authors, there is still applicability to scaling MCMC for large samples. Beyond pseudo-marginal situations.)

“…this, at first sight very disappointing behavior of the LMC algorithm is, in fact, continuously connected to the exponential convergence of the gradient descent.”

The paper concludes with an interesting mise en parallèle of Langevin algorithms and of gradient descent algorithms, since the convergence rates are the same.