**T**oday is the first day of the FUSION workshop Rémi Bardenet and myself organised. Due to schedule clashes, I will alas not be there, since [no alas!] at the BNP conference in Chili. The program and collection of participants is quite exciting and I hope more fusion will result from this meeting. Enjoy! (And beware of boars, cold water, and cliffs!!!)

## Archive for SMC

## Fusion at CIRM

Posted in Mountains, pictures, Statistics, Travel, University life with tags ABC, Bayesian non-parametrics, BNP, boar, Chili, CIRM, cold water swimming, data privacy, fusion, Les Calanques, Luminy, Luminy campus, Méditerranée, MCMC, Parc National des Calanques, particle filter, SMC, Université Aix Marseille, workshop on October 24, 2022 by xi'an## evidence estimation in finite and infinite mixture models

Posted in Books, Statistics, University life with tags arXiv, Bayes factor, Bayesian model evaluation, bridge sampling, candidate's formula, Charlie Geyer, Chib's approximation, Dirichlet process mixture, nested sampling, noise contrasting estimation, sequential Monte Carlo, SIS, SMC, Université Paris Dauphine on May 20, 2022 by xi'an**A**drien Hairault (PhD student at Dauphine), Judith and I just arXived a new paper on evidence estimation for mixtures. This may sound like a well-trodden path that I have repeatedly explored in the past, but methinks that estimating the model evidence doth remain a notoriously difficult task for large sample or many component finite mixtures and even more for “infinite” mixture models corresponding to a Dirichlet process. When considering different Monte Carlo techniques advocated in the past, like Chib’s (1995) method, SMC, or bridge sampling, they exhibit a range of performances, in terms of computing time… One novel (?) approach in the paper is to write Chib’s (1995) identity for partitions rather than parameters as (a) it bypasses the label switching issue (as we already noted in Hurn et al., 2000), another one is to exploit Geyer (1991-1994) reverse logistic regression technique in the more challenging Dirichlet mixture setting, and yet another one a sequential importance sampling solution à la Kong et al. (1994), as also noticed by Carvalho et al. (2010). [We did not cover nested sampling as it quickly becomes onerous.]

Applications are numerous. In particular, testing for the number of components in a finite mixture model or against the fit of a finite mixture model for a given dataset has long been and still is an issue of much interest and diverging opinions, albeit yet missing a fully satisfactory resolution. Using a Bayes factor to find the right number of components K in a finite mixture model is known to provide a consistent procedure. We furthermore establish there the consistence of the Bayes factor when comparing a parametric family of finite mixtures against the nonparametric ‘strongly identifiable’ Dirichlet Process Mixture (DPM) model.

## living on the edge [of the canal]

Posted in Books, pictures, Statistics, Travel, University life with tags arXiv, canals, Cannaregio Canal, Hammersley, Latin hypercube, MCMC, qMC, Reuven Rubinstein, SMC, stochastic volatility, Università Ca' Foscari Venezia, Venezia on December 15, 2021 by xi'an**L**ast month, Roberto Casarin, Radu Craiu, Lorenzo Frattarolo and myself posted an arXiv paper on a unified approach to antithetic sampling. To which I mostly and modestly contributed while visiting Roberto in Venezia two years ago (although it seems much farther than that!). I have always found antithetic sampling fascinating, albeit mostly unachievable in realistic situations, except (and approximately) by quasi-random tools. The original approach dates back to Hammersley and Morton, circa 1956, when they optimally couple X=F⁻(U) and Y=F⁻(1-U), with U Uniform, although there is no clear-cut extension beyond pairs or above dimension one. While the search for optimal and feasible antithetic plans dried out in the mid-1980’s, despite near successes by Rubinstein and others, the focus switched to Latin hypercube sampling.

The construction of a general antithetic sampling scheme is based on sampling uniformly an edge within an undirected graph in the d-dimensional hypercube, under some (three) assumptions on the edges to achieve uniformity for the marginals. This construction achieves the smallest Kullback-Leibler divergence between the resulting joint and the product of uniforms. And it can be furthermore constrained to be d-countermonotonic, ie such that a non-linear sum of the components is constant. We also show that the proposal leads to closed-form Kendall’s τ and Spearman’s ρ. Which can be used to assess different d-countermonotonic schemes, incl. earlier ones found in the literature. The antithetic sampling proposal can be applied in Monte Carlo, Markov chain Monte Carlo, and sequential Monte Carlo settings. In a stochastic volatility example of the later (SMC) we achieve performances similar to the quasi-Monte Carlo approach of Mathieu Gerber and Nicolas Chopin.

## ordered allocation sampler

Posted in Books, Statistics with tags Data augmentation, Galaxy, Gibbs sampling, hidden Markov models, JASA, label switching, latent variable models, MCMC, partition function, random partition trees, SMC, statistical methodology on November 29, 2021 by xi'an**R**ecently, 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)!