Archive for MCMC

ordered allocation sampler

Posted in Books, Statistics with tags , , , , , , , , , , , on November 29, 2021 by xi'an

Recently, Pierpaolo De Blasi and María Gil-Leyva arXived a proposal for a novel Gibbs sampler for mixture models. In both finite and infinite mixture models. In connection with Pitman (1996) theory of species sampling and with interesting features in terms of removing the vexing label switching features.

The key idea is to work with the mixture components in the random order of appearance in an exchangeable sequence from the mixing distribution (…) In accordance with the order of appearance, we derive a new Gibbs sampling algorithm that we name the ordered allocation sampler. “

This central idea is thus a reinterpretation of the mixture model as the marginal of the component model when its parameter is distributed as a species sampling variate. An ensuing marginal algorithm is to integrate out the weights and the allocation variables to only consider the non-empty component parameters and the partition function, which are label invariant. Which reminded me of the proposal we made in our 2000 JASA paper with Gilles Celeux and Merrilee Hurn (one of my favourite papers!). And of the [first paper in Statistical Methodology] 2004 partitioned importance sampling version with George Casella and Marty Wells. As in the later, the solution seems to require the prior on the component parameters to be conjugate (as I do not see a way to produce an unbiased estimator of the partition allocation probabilities).

The ordered allocation sample considers the posterior distribution of the different object made of the parameters and of the sequence of allocations to the components for the sample written in a given order, ie y¹,y², &tc. Hence y¹ always gets associated with component 1, y² with either component 1 or component 2, and so on. For this distribution, the full conditionals are available, incl. the full posterior on the number m of components, only depending on the data through the partition sizes and the number m⁺ of non-empty components. (Which relates to the debate as to whether or not m is estimable…) This sequential allocation reminded me as well of an earlier 2007 JRSS paper by Nicolas Chopin. Albeit using particles rather than Gibbs and applied to a hidden Markov model. Funny enough, their synthetic dataset univ4 almost resembles the Galaxy dataset (as in the above picture of mine)!

control variates [seminar]

Posted in pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , , on November 5, 2021 by xi'an

Today, Petros Dellaportas (whom I have know since the early days of MCMC, when we met in CIRM) gave a seminar at the Warwick algorithm seminar on control variates for MCMC, reminding me of his 2012 JRSS paper. Based on the Poisson equation and using a second control variate to stabilise the Monte Carlo approximation do the first control variate. The difference with usual control variates is finding a first approximate G(x)-q(y|x)G(Y) to F-πF. And the first Poisson equation is using α(x,y)q(y|x) rather than π. Then the second expands log α(x,y)q(y|x) to achieve a manageable term.

Abstract: We provide a general methodology to construct control variates for any discrete time random walk Metropolis and Metropolis-adjusted Langevin algorithm Markov chains that can achieve, in a post-processing manner and with a negligible additional computational cost, impressive variance reduction when compared to the standard MCMC ergodic averages. Our proposed estimators are based on an approximate solution of the Poisson equation for a multivariate Gaussian target densities of any dimension.

I wonder if there were a neural network version that would first build G from scratch and later optimise it towards solving the Poisson equation. As in this recent arXival I haven’t read (yet).

Blackwell-Rosenbluth Awards 2021

Posted in Statistics, University life with tags , , , , , , , , , , , on November 1, 2021 by xi'an

Congratulations to the winners of the newly created award! This j-ISBA award is intended for junior researchers in different areas of Bayesian statistics. And named after David Blackwell and Arianna  Rosenbluth. They will present their work at the newly created JB³ seminars on 10 and 12 November, both at 1pm UTC. (The awards are broken into two time zones, corresponding to the Americas and the rest of the World.)

UTC+0 to UTC+13

Marta Catalano, Warwick University
Samuel Livingstone, University College London
Dootika Vats, Indian Institute of Technology Kanpur

UTC-12 to UTC-1

Trevor Campbell, University of British Columbia
Daniel Kowal, Rice University
Yixin Wang, University of Michigan

Basque thesis defence [Bayes almost on the beach]

Posted in Books, Kids, pictures, Statistics, Travel, University life with tags , , , , , , , , , , , , , , , on October 21, 2021 by xi'an

Yesterday morning I took part in a thesis defence (as a jury member) in the coastal city of Anglet, in the (French part of the) Basque Country. The PhD candidate was Sébastien Coube-Sisqueille, whom I did not know directly (although we had crossed paths at CIRM years ago and he had attended my MCMC course at ENSAE even more years ago). As it happened all other members of the committee, apart from Sébastien’s advisor, Benoît Liquet, were on Teams, being unable to travel to the Basque Country. Sébastien’s thesis is about MCMC strategies to accelerate convergence in spatial models represented as nearest neighbor Gaussian processes (NNGP), which relates to the earlier works of (X)XL on interweaving. (Unsurprisingly, the defence was successful and the candidate awarded his PhD!) Icing on the cake, I managed to take a dip in the Atlantic Ocean, before flying back to Paris for dinner, on a very warm afternoon (and slightly cooler water), thanks to Sébastien driving me to a nearby beach!

GANs as density estimators

Posted in Books, Statistics with tags , , , , , , , on October 15, 2021 by xi'an

I recently read an arXival entitled Conditional Sampling With Monotone GAN by Kovakchi et al., who construct  a mapping T that transforms or pushes forward a reference measure þ() like a multivariate Normal distribution to a target conditional distribution ð(dθ|x).  Which makes the proposal a type of normalising flow, except it does not require a Jacobian derivation… The mapping T is monotonous and block triangular in order to be invertible. It is learned from data by minimising a functional divergence between Tþ(dθ) and ð(dθ|x), for instance GAN least square or GAN Wasserstein penalties and representing T as a neural network.  Where monotonicity is imposed by a Lagrangian. The authors “note that global minimizers of [their GAN criterion] can also be used for conditional density estimation” but I fail to understand the distinction in that once T is constructed, the estimated conditional density is automatically available. However my main source of puzzlement is at the worth of this construction, since it does not provide an exact generative process for the conditional distribution, while requiring many generations from the joint distribution. Rather than a comparison with MCMC, which is not applicable in untractable generative models, a comparison with less expensive ABC solutions would have been appropriate, I think. And the paper is missing any quantification on the quality or asymptotics of the density estimate provided by this involved approximation, as most of the recent literature on normalising flows and friends. (A point acknowledged by the authors in the supplementary material section.)

“In this regard, the MGANs approach introduced in the article belongs to the category of sampling techniques such as MCMC, whose goal is to generate independent samples from the law of y|x, as opposed to assuming some structural form of the probability measure directly.”

I am unsure I understand the above remark as MCMC methods are intrinsically linked with the exact probability distribution, exploiting either some conditional representations as in Gibbs or at the very least the ability to compute the joint density…

 

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