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Judith Rousseau

Publications and source records attributed to Judith Rousseau.

At least 55 records · Page 3Linked to original sources
arXiv2018

Nonparametric Bayesian estimation of multivariate Hawkes processes

This paper studies nonparametric estimation of parameters of multivariate Hawkes processes. We consider the Bayesian setting and derive posterior concentration rates. First rates are derived for L1-metrics for stochastic intensities of the Hawkes process. We then deduce rates for the L1-norm of interactions functions of the process. Our results are exemplified by using priors based on piecewise constant functions, with regular or random partitions and priors based on mixtures of Betas distributions. Numerical illustrations are then proposed with in mind applications for inferring functional connec-tivity graphs of neurons.

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arXiv2017

Efficient semiparametric estimation and model selection for multidimensional mixtures

In this paper, we consider nonparametric multidimensional finite mixture models and we are interested in the semiparametric estimation of the population weights. Here, the i.i.d. observations are assumed to have at least three components which are independent given the population. We approximate the semiparametric model by projecting the conditional distributions on step functions associated to some partition. Our first main result is that if we refine the partition slowly enough, the associated sequence of maximum likelihood estimators of the weights is asymptotically efficient, and the posterior distribution of the weights, when using a Bayesian procedure, satisfies a semiparametric Bernstein von Mises theorem. We then propose a cross-validation like procedure to select the partition in a finite horizon. Our second main result is that the proposed procedure satisfies an oracle inequality. Numerical experiments on simulated data illustrate our theoretical results.

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arXiv2017

Adaptive density estimation based on a mixture of Gammas

We consider the problem of Bayesian density estimation on the positive semiline for possibly unbounded densities. We propose a hierarchical Bayesian estimator based on the gamma mixture prior which can be viewed as a location mixture. We study convergence rates of Bayesian density estimators based on such mixtures. We construct approximations of the local Hölder densities, and of their extension to unbounded densities, to be continuous mixtures of gamma distributions, leading to approximations of such densities by finite mixtures. These results are then used to derive posterior concentration rates, with priors based on these mixture models. The rates are minimax (up to a log n term) and since the priors are independent of the smoothness the rates are adaptive to the smoothness.

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arXiv2016

Some comments about "Penalising model component complexity" by Simpson et al. (2017)

This note discusses the paper "Penalising model component complexity" by Simpson et al. (2017). While we acknowledge the highly novel approach to prior construction and commend the authors for setting new-encompassing principles that will Bayesian modelling, and while we perceive the potential connection with other branches of the literature, we remain uncertain as to what extent the principles exposed in the paper can be developed outside specific models, given their lack of precision. The very notions of model component, base model, overfitting prior are for instance conceptual rather than mathematical and we thus fear the concept of penalised complexity may not further than extending first-guess priors into larger families, thus failing to establish reference priors on a novel sound ground.

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arXiv2016

Tails assumptions and posterior concentration rates for mixtures of Gaussians

Nowadays in density estimation, posterior rates of convergence for location and location-scale mixtures of Gaussians are only known under light-tail assumptions; with better rates achieved by location mixtures. It is conjectured, but not proved, that the situation should be reversed under heavy tails assumptions. The conjecture is based on the feeling that there is no need to achieve a good order of approximation in regions with few data (say, in the tails), favoring location-scale mixtures which allow for spatially varying order of approximation. Here we test the previous argument on the Gaussian errors mean regression model with random design, for which the light tail assumption is not required for proofs. Although we cannot invalidate the conjecture due to the lack of lower bound, we find that even with heavy tails assumptions, location-scale mixtures apparently perform always worst than location mixtures. However, the proofs suggest to introduce hybrid location-scale mixtures that are find to outperform both location and location-scale mixtures, whatever the nature of the tails. Finally, we show that all tails assumptions can be released at the price of making the prior distribution covariate dependent.

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arXiv2016

Some comments about James Watson's and Chris Holmes' "Approximate Models and Robust Decisions": Nonparametric Bayesian clay for robust decision bricks

This note discusses Watson and Holmes (2016) and their pro- posals towards more robust Bayesian decisions. While we acknowledge and commend the authors for setting new and all-encompassing prin- ciples of Bayesian robustness, and we appreciate the strong anchoring of those within a decision-theoretic referential, we remain uncertain as to which extent such principles can be applied outside binary de- cisions. We also wonder at the ultimate relevance of Kullback-Leibler neighbourhoods to characterise robustness and favour extensions along non-parametric axes.

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arXiv2016

Asymptotic behaviour of the empirical Bayes posteriors associated to maximum marginal likelihood estimator

We consider the asymptotic behaviour of the marginal maximum likelihood empirical Bayes posterior distribution in general setting. First we characterize the set where the maximum marginal likelihood estimator is located with high probability. Then we provide oracle type of upper and lower bounds for the contraction rates of the empirical Bayes posterior. We also show that the hierarchical Bayes posterior achieves the same contraction rate as the maximum marginal likelihood empirical Bayes posterior. We demonstrate the applicability of our general results for various models and prior distributions by deriving upper and lower bounds for the contraction rates of the corresponding empirical and hierarchical Bayes posterior distributions.

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arXiv2016

Overfitting hidden Markov models with an unknown number of states

This paper presents new theory and methodology for the Bayesian estimation of overfitted hidden Markov models, with finite state space. The goal is then to achieve posterior emptying of extra states. A prior configuration is constructed which favours configurations where the hidden Markov chain remains ergodic although it empties out some of the states. Asymptotic posterior convergence rates are proven theoretically, and demonstrated with a large sample simulation. The problem of overfitted HMMs is then considered in the context of smaller sample sizes, and due to computational and mixing issues two alternative prior structures are studied, one commonly used in practice, and a mixture of the two priors. The Prior Parallel Tempering approach of van Havre (2015) is also extended to HMMs to allow MCMC estimation of the complex posterior space. A replicate simulation study and an in-depth exploration is performed to compare the three priors with hyperparameters chosen according to the asymptotic constraints alongside less informative alternatives.

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arXiv2016

Clustering action potential spikes: Insights on the use of overfitted finite mixture models and Dirichlet process mixture models

The modelling of action potentials from extracellular recordings, or spike sorting, is a rich area of neuroscience research in which latent variable models are often used. Two such models, Overfitted Finite Mixture models (OFMs) and Dirichlet Process Mixture models (DPMs) are considered to provide insights for unsupervised clustering of complex, multivariate medical data when the number of clusters is unknown. OFM and DPM are structured in a similar hierarchical fashion but they are based on different philosophies with different underlying assumptions. This study investigates how these differences impact on a real study of spike sorting, for the estimation of multivariate Gaussian location-scale mixture models in the presence of common difficulties arising from complex medical data. The results provide insights allowing the future analyst to choose an approach suited to the situation and goal of the research problem at hand.

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arXiv2015

A Bernstein-von Mises theorem for smooth functionals in semiparametric models

A Bernstein-von Mises theorem is derived for general semiparametric functionals. The result is applied to a variety of semiparametric problems in i.i.d. and non-i.i.d. situations. In particular, new tools are developed to handle semiparametric bias, in particular for nonlinear functionals and in cases where regularity is possibly low. Examples include the squared $L^2$-norm in Gaussian white noise, nonlinear functionals in density estimation, as well as functionals in autoregressive models. For density estimation, a systematic study of BvM results for two important classes of priors is provided, namely random histograms and Gaussian process priors.

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arXiv2015

On adaptive posterior concentration rates

We investigate the problem of deriving posterior concentration rates under different loss functions in nonparametric Bayes. We first provide a lower bound on posterior coverages of shrinking neighbourhoods that relates the metric or loss under which the shrinking neighbourhood is considered, and an intrinsic pre-metric linked to frequentist separation rates. In the Gaussian white noise model, we construct feasible priors based on a spike and slab procedure reminiscent of wavelet thresholding that achieve adaptive rates of contraction under $L^2$ or $L^{\infty}$ metrics when the underlying parameter belongs to a collection of Hölder balls and that moreover achieve our lower bound. We analyse the consequences in terms of asymptotic behaviour of posterior credible balls as well as frequentist minimax adaptive estimation. Our results are appended with an upper bound for the contraction rate under an arbitrary loss in a generic regular experiment. The upper bound is attained for certain sieve priors and enables to extend our results to density estimation.

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arXiv2015

Overfitting Bayesian Mixture Models with an Unknown Number of Components

This paper proposes solutions to three issues pertaining to the estimation of finite mixture models with an unknown number of components: the non-identifiability induced by overfitting the number of components, the mixing limitations of standard Markov Chain Monte Carlo (MCMC) sampling techniques, and the related label switching problem. An overfitting approach is used to estimate the number of components in a finite mixture model via a Zmix algorithm. Zmix provides a bridge between multidimensional samplers and test based estimation methods, whereby priors are chosen to encourage extra groups to have weights approaching zero. MCMC sampling is made possible by the implementation of prior parallel tempering, an extension of parallel tempering. Zmix can accurately estimate the number of components, posterior parameter estimates and allocation probabilities given a sufficiently large sample size. The results will reflect uncertainty in the final model and will report the range of possible candidate models and their respective estimated probabilities from a single run. Label switching is resolved with a computationally light-weight method, Zswitch, developed for overfitted mixtures by exploiting the intuitiveness of allocation-based relabelling algorithms and the precision of label-invariant loss functions. Four simulation studies are included to illustrate Zmix and Zswitch, as well as three case studies from the literature. All methods are available as part of the R package Zmix, which can currently be applied to univariate Gaussian mixture models

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arXiv2015

Bayesian nonparametric dependent model for partially replicated data: the influence of fuel spills on species diversity

We introduce a dependent Bayesian nonparametric model for the probabilistic modeling of membership of subgroups in a community based on partially replicated data. The focus here is on species-by-site data, i.e. community data where observations at different sites are classified in distinct species. Our aim is to study the impact of additional covariates, for instance environmental variables, on the data structure, and in particular on the community diversity. To that purpose, we introduce dependence a priori across the covariates, and show that it improves posterior inference. We use a dependent version of the Griffiths-Engen-McCloskey distribution defined via the stick-breaking construction. This distribution is obtained by transforming a Gaussian process whose covariance function controls the desired dependence. The resulting posterior distribution is sampled by Markov chain Monte Carlo. We illustrate the application of our model to a soil microbial dataset acquired across a hydrocarbon contamination gradient at the site of a fuel spill in Antarctica. This method allows for inference on a number of quantities of interest in ecotoxicology, such as diversity or effective concentrations, and is broadly applicable to the general problem of communities response to environmental variables.

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arXiv2014

About the posterior distribution in hidden Markov models with unknown number of states

We consider finite state space stationary hidden Markov models (HMMs) in the situation where the number of hidden states is unknown. We provide a frequentist asymptotic evaluation of Bayesian analysis methods. Our main result gives posterior concentration rates for the marginal densities, that is for the density of a fixed number of consecutive observations. Using conditions on the prior, we are then able to define a consistent Bayesian estimator of the number of hidden states. It is known that the likelihood ratio test statistic for overfitted HMMs has a nonstandard behaviour and is unbounded. Our conditions on the prior may be seen as a way to penalize parameters to avoid this phenomenon. Inference of parameters is a much more difficult task than inference of marginal densities, we still provide a precise description of the situation when the observations are i.i.d. and we allow for $2$ possible hidden states.

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arXiv2014

Bayesian matrix completion: prior specification

Low-rank matrix estimation from incomplete measurements recently received increased attention due to the emergence of several challenging applications, such as recommender systems; see in particular the famous Netflix challenge. While the behaviour of algorithms based on nuclear norm minimization is now well understood, an as yet unexplored avenue of research is the behaviour of Bayesian algorithms in this context. In this paper, we briefly review the priors used in the Bayesian literature for matrix completion. A standard approach is to assign an inverse gamma prior to the singular values of a certain singular value decomposition of the matrix of interest; this prior is conjugate. However, we show that two other types of priors (again for the singular values) may be conjugate for this model: a gamma prior, and a discrete prior. Conjugacy is very convenient, as it makes it possible to implement either Gibbs sampling or Variational Bayes. Interestingly enough, the maximum a posteriori for these different priors is related to the nuclear norm minimization problems. We also compare all these priors on simulated datasets, and on the classical MovieLens and Netflix datasets.

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