arXiv subjects

Judith Rousseau

Publications and source records attributed to Judith Rousseau.

72 records · Page 4Linked to original sources

arXiv2014

Posterior concentration rates for counting processes with Aalen multiplicative intensities

We provide general conditions to derive posterior concentration rates for Aalen counting processes. The conditions are designed to resemble those proposed in the literature for the problem of density estimation, for instance in Ghosal et al. (2000), so that existing results on density estimation can be adapted to the present setting. We apply the general theorem to some prior models including Dirichlet process mixtures of uniform densities to estimate monotone non-increasing intensities and log-splines.

stat.ME↗

arXiv2014

Posterior concentration rates for empirical Bayes procedures, with applications to Dirichlet Process mixtures

In this paper we provide general conditions to check on the model and the prior to derive posterior concentration rates for data-dependent priors (or empirical Bayes approaches). We aim at providing conditions that are close to the conditions provided in the seminal paper by Ghosal and van der Vaart (2007a). We then apply the general theorem to two different settings: the estimation of a density using Dirichlet process mixtures of Gaussian random variables with base measure depending on some empirical quantities and the estimation of the intensity of a counting process under the Aalen model. A simulation study for inhomogeneous Poisson processes also illustrates our results. In the former case we also derive some results on the estimation of the mixing density and on the deconvolution problem. In the latter, we provide a general theorem on posterior concentration rates for counting processes with Aalen multiplicative intensity with priors not depending on the data.

math.ST↗

arXiv2014

Using informative priors in the estimation of mixtures over time with application to aerosol particle size distributions

The issue of using informative priors for estimation of mixtures at multiple time points is examined. Several different informative priors and an independent prior are compared using samples of actual and simulated aerosol particle size distribution (PSD) data. Measurements of aerosol PSDs refer to the concentration of aerosol particles in terms of their size, which is typically multimodal in nature and collected at frequent time intervals. The use of informative priors is found to better identify component parameters at each time point and more clearly establish patterns in the parameters over time. Some caveats to this finding are discussed.

stat.AP↗

arXiv2013

Bayesian optimal adaptive estimation using a sieve prior

We derive rates of contraction of posterior distributions on nonparametric models resulting from sieve priors. The aim of the paper is to provide general conditions to get posterior rates when the parameter space has a general structure, and rate adaptation when the parameter space is, e.g., a Sobolev class. The conditions employed, although standard in the literature, are combined in a different way. The results are applied to density, regression, nonlinear autoregression and Gaussian white noise models. In the latter we have also considered a loss function which is different from the usual l2 norm, namely the pointwise loss. In this case it is possible to prove that the adaptive Bayesian approach for the l2 loss is strongly suboptimal and we provide a lower bound on the rate.

math.ST↗

arXiv2013

Non parametric finite translation mixtures with dependent regime

In this paper we consider non parametric finite translation mixtures. We prove that all the parameters of the model are identifiable as soon as the matrix that defines the joint distribution of two consecutive latent variables is non singular and the translation parameters are distinct. Under this assumption, we provide a consistent estimator of the number of populations, of the translation parameters and of the distribution of two consecutive latent variables, which we prove to be asymptotically normally distributed under mild dependency assumptions. We propose a non parametric estimator of the unknown translated density. In case the latent variables form a Markov chain (Hidden Markov models), we prove an oracle inequality leading to the fact that this estimator is minimax adaptive over regularity classes of densities.

math.ST↗

arXiv2012

Computational aspects of Bayesian spectral density estimation

Gaussian time-series models are often specified through their spectral density. Such models present several computational challenges, in particular because of the non-sparse nature of the covariance matrix. We derive a fast approximation of the likelihood for such models. We propose to sample from the approximate posterior (that is, the prior times the approximate likelihood), and then to recover the exact posterior through importance sampling. We show that the variance of the importance sampling weights vanishes as the sample size goes to infinity. We explain why the approximate posterior may typically multi-modal, and we derive a Sequential Monte Carlo sampler based on an annealing sequence in order to sample from that target distribution. Performance of the overall approach is evaluated on simulated and real datasets. In addition, for one real world dataset, we provide some numerical evidence that a Bayesian approach to semi-parametric estimation of spectral density may provide more reasonable results than its Frequentist counter-parts.

stat.CO↗

arXiv2012

Bayesian nonparametric estimation of the spectral density of a long or intermediate memory Gaussian process

A stationary Gaussian process is said to be long-range dependent (resp., anti-persistent) if its spectral density $f(λ)$ can be written as $f(λ)=|λ|^{-2d}g(|λ|)$, where $0<d<1/2$ (resp., $-1/2<d<0$), and $g$ is continuous and positive. We propose a novel Bayesian nonparametric approach for the estimation of the spectral density of such processes. We prove posterior consistency for both $d$ and $g$, under appropriate conditions on the prior distribution. We establish the rate of convergence for a general class of priors and apply our results to the family of fractionally exponential priors. Our approach is based on the true likelihood and does not resort to Whittle's approximation.

stat.ME↗

arXiv2012

Bayes and empirical Bayes: do they merge?

Bayesian inference is attractive for its coherence and good frequentist properties. However, it is a common experience that eliciting a honest prior may be difficult and, in practice, people often take an {\em empirical Bayes} approach, plugging empirical estimates of the prior hyperparameters into the posterior distribution. Even if not rigorously justified, the underlying idea is that, when the sample size is large, empirical Bayes leads to "similar" inferential answers. Yet, precise mathematical results seem to be missing. In this work, we give a more rigorous justification in terms of merging of Bayes and empirical Bayes posterior distributions. We consider two notions of merging: Bayesian weak merging and frequentist merging in total variation. Since weak merging is related to consistency, we provide sufficient conditions for consistency of empirical Bayes posteriors. Also, we show that, under regularity conditions, the empirical Bayes procedure asymptotically selects the value of the hyperparameter for which the prior mostly favors the "truth". Examples include empirical Bayes density estimation with Dirichlet process mixtures.

math.ST↗

arXiv2012

Bayesian semi-parametric estimation of the long-memory parameter under FEXP-priors

For a Gaussian time series with long-memory behavior, we use the FEXP-model for semi-parametric estimation of the long-memory parameter $d$. The true spectral density $f_o$ is assumed to have long-memory parameter $d_o$ and a FEXP-expansion of Sobolev-regularity $\be > 1$. We prove that when $k$ follows a Poisson or geometric prior, or a sieve prior increasing at rate $n^{\frac{1}{1+2\be}}$, $d$ converges to $d_o$ at a suboptimal rate. When the sieve prior increases at rate $n^{\frac{1}{2\be}}$ however, the minimax rate is almost obtained. Our results can be seen as a Bayesian equivalent of the result which Moulines and Soulier obtained for some frequentist estimators.

math.ST↗

arXiv2011

Inherent Difficulties of Non-Bayesian Likelihood-based Inference, as Revealed by an Examination of a Recent Book by Aitkin

For many decades, statisticians have made attempts to prepare the Bayesian omelette without breaking the Bayesian eggs; that is, to obtain probabilistic likelihood-based inferences without relying on informative prior distributions. A recent example is Murray Aitkin's recent book, {\em Statistical Inference}, which presents an approach to statistical hypothesis testing based on comparisons of posterior distributions of likelihoods under competing models. Aitkin develops and illustrates his method using some simple examples of inference from iid data and two-way tests of independence. We analyze in this note some consequences of the inferential paradigm adopted therein, discussing why the approach is incompatible with a Bayesian perspective and why we do not find it relevant for applied work.

stat.ME↗

arXiv2010

Bayesian Inference

This chapter provides a overview of Bayesian inference, mostly emphasising that it is a universal method for summarising uncertainty and making estimates and predictions using probability statements conditional on observed data and an assumed model (Gelman 2008). The Bayesian perspective is thus applicable to all aspects of statistical inference, while being open to the incorporation of information items resulting from earlier experiments and from expert opinions. We provide here the basic elements of Bayesian analysis when considered for standard models, refering to Marin and Robert (2007) and to Robert (2007) for book-length entries.1 In the following, we refrain from embarking upon philosophical discussions about the nature of knowledge (see, e.g., Robert 2007, Chapter 10), opting instead for a mathematically sound presentation of an eminently practical statistical methodology. We indeed believe that the most convincing arguments for adopting a Bayesian version of data analyses are in the versatility of this tool and in the large range of existing applications, rather than in those polemical arguments.

stat.ME↗

arXiv2010

On Bayesian Data Analysis

This introduction to Bayesian statistics presents the main concepts as well as the principal reasons advocated in favour of a Bayesian modelling. We cover the various approaches to prior determination as well as the basis asymptotic arguments in favour of using Bayes estimators. The testing aspects of Bayesian inference are also examined in details.

stat.ME↗

arXiv2010

Harold Jeffreys's Theory of Probability Revisited

Published exactly seventy years ago, Jeffreys's Theory of Probability (1939) has had a unique impact on the Bayesian community and is now considered to be one of the main classics in Bayesian Statistics as well as the initiator of the objective Bayes school. In particular, its advances on the derivation of noninformative priors as well as on the scaling of Bayes factors have had a lasting impact on the field. However, the book reflects the characteristics of the time, especially in terms of mathematical rigor. In this paper we point out the fundamental aspects of this reference work, especially the thorough coverage of testing problems and the construction of both estimation and testing noninformative priors based on functional divergences. Our major aim here is to help modern readers in navigating in this difficult text and in concentrating on passages that are still relevant today.

math.ST↗

arXiv2010

Rejoinder: Harold Jeffreys's Theory of Probability Revisited

We are grateful to all discussants of our re-visitation for their strong support in our enterprise and for their overall agreement with our perspective. Further discussions with them and other leading statisticians showed that the legacy of Theory of Probability is alive and lasting. [arXiv:0804.3173]

stat.ME↗

arXiv2010

Rates of convergence for the posterior distributions of mixtures of Betas and adaptive nonparametric estimation of the density

In this paper, we investigate the asymptotic properties of nonparametric Bayesian mixtures of Betas for estimating a smooth density on $[0,1]$. We consider a parametrization of Beta distributions in terms of mean and scale parameters and construct a mixture of these Betas in the mean parameter, while putting a prior on this scaling parameter. We prove that such Bayesian nonparametric models have good frequentist asymptotic properties. We determine the posterior rate of concentration around the true density and prove that it is the minimax rate of concentration when the true density belongs to a Hölder class with regularity $β$, for all positive $β$, leading to a minimax adaptive estimating procedure of the density. We also believe that the approximating results obtained on these mixtures of Beta densities can be of interest in a frequentist framework.

math.ST↗

arXiv2009

Bernstein Von Mises Theorem for linear functionals of the density

In this paper, we study the asymptotic posterior distribution of linear functionals of the density. In particular, we give general conditions to obtain a semiparametric version of the Bernstein-Von Mises theorem. We then apply this general result to nonparametric priors based on infinite dimensional exponential families. As a byproduct, we also derive adaptive nonparametric rates of concentration of the posterior distributions under these families of priors on the class of Sobolev and Besov spaces.

math.ST↗

arXiv2008

Bounds for Bayesian order identification with application to mixtures

The efficiency of two Bayesian order estimators is studied. By using nonparametric techniques, we prove new underestimation and overestimation bounds. The results apply to various models, including mixture models. In this case, the errors are shown to be $O(e^{-an})$ and $O((\log n)^b/\sqrt{n})$ ($a,b>0$), respectively.

math.ST↗

arXiv2007

Bayesian nonparametric estimation of the spectral density of a long memory Gaussian time series

Let $\mathbf {X}=\{X_t, t=1,2,... \}$ be a stationary Gaussian random process, with mean $EX_t=μ$ and covariance function $γ(τ)=E(X_t-μ)(X_{t+τ}-μ)$. Let $f(λ)$ be the corresponding spectral density; a stationary Gaussian process is said to be long-range dependent, if the spectral density $f(λ)$ can be written as the product of a slowly varying function $\tilde{f}(λ)$ and the quantity $λ^{-2d}$. In this paper we propose a novel Bayesian nonparametric approach to the estimation of the spectral density of $\mathbf {X}$. We prove that, under some specific assumptions on the prior distribution, our approach assures posterior consistency both when $f(\cdot)$ and $d$ are the objects of interest. The rate of convergence of the posterior sequence depends in a significant way on the structure of the prior; we provide some general results and also consider the fractionally exponential (FEXP) family of priors (see below). Since it has not a well founded justification in the long memory set-up, we avoid using the Whittle approximation to the likelihood function and prefer to use the true Gaussian likelihood.

math.ST↗