13 research outputs found
Besov-Laplace priors in density estimation: optimal posterior contraction rates and adaptation
Besov priors are nonparametric priors that model spatially inhomogeneous
functions. They are routinely used in inverse problems and imaging, where they
exhibit attractive sparsity-promoting and edge-preserving features. A recent
line of work has initiated the study of their asymptotic frequentist
convergence properties. In the present paper, we consider the theoretical
recovery performance of the posterior distributions associated to Besov-Laplace
priors in the density estimation model, under the assumption that the
observations are generated by a possibly spatially inhomogeneous true density
belonging to a Besov space. We improve on existing results and show that
carefully tuned Besov-Laplace priors attain optimal posterior contraction
rates. Furthermore, we show that hierarchical procedures involving a
hyper-prior on the regularity parameter lead to adaptation to any smoothness
level.Comment: 35 page
Inférence non-paramétrique pour des interactions poissoniennes
L'objet de cette thèse est d'étudier divers problèmes de statistique non-paramétrique dans le cadre d'un modèle d'interactions poissoniennes. De tels modèles sont, par exemple, utilisés en neurosciences pour analyser les interactions entre deux neurones au travers leur émission de potentiels d'action au cours de l'enregistrement de l'activité cérébrale ou encore en génomique pour étudier les distances favorisées ou évitées entre deux motifs le long du génome. Dans ce cadre, nous introduisons une fonction dite de reproduction qui permet de quantifier les positions préférentielles des motifs et qui peut être modélisée par l'intensité d'un processus de Poisson. Dans un premier temps, nous nous intéressons à l'estimation de cette fonction que l'on suppose très localisée. Nous proposons une procédure d'estimation adaptative par seuillage de coefficients d'ondelettes qui est optimale des points de vue oracle et minimax. Des simulations et une application en génomique sur des données réelles provenant de la bactérie E. coli nous permettent de montrer le bon comportement pratique de notre procédure. Puis, nous traitons les problèmes de test associés qui consistent à tester la nullité de la fonction de reproduction. Pour cela, nous construisons une procédure de test optimale du point de vue minimax sur des espaces de Besov faibles, qui a également montré ses performances du point de vue pratique. Enfin, nous prolongeons ces travaux par l'étude d'une version discrète en grande dimension du modèle précédent en proposant une procédure adaptative de type Lasso.The subject of this thesis is the study of some adaptive nonparametric statistical problems in the framework of a Poisson interactions model. Such models are used, for instance, in neurosciences to analyze interactions between two neurons through their spikes emission during the recording of the brain activity or in genomics to study favored or avoided distances between two motifs along a genome. In this setting, we naturally introduce a so-called reproduction function that allows to quantify the favored positions of the motifs and which is considered as the intensity of a Poisson process. Our first interest is the estimation of this function assumed to be well localized. We propose a data-driven wavelet thresholding estimation procedure that is optimal from oracle and minimax points of view. Simulations and an application to genomic data from the bacterium E. coli allow us to show the good practical behavior of our procedure. Then, we deal with associated problems on tests which consist in testing the nullity of the reproduction function. For this purpose, we build a minimax optimal testing procedure on weak Besov spaces and we provide some simulations showing good practical performances of our procedure. Finally, we extend this work with the study of a high-dimensional discrete setting of our previous model by proposing an adaptive Lasso-type procedure.PARIS11-SCD-Bib. électronique (914719901) / SudocSudocFranceF
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Asymptotic theory for Bayesian nonparametric inference in statistical models arising from partial differential equations
Partial differential equations (PDEs) are primary mathematical tools to model the behaviour of complex real-world systems. PDEs generally include a collection of parameters in their formulation, which are often unknown in applications and need to be estimated from the data. In the present thesis, we investigate the theoretical performance of nonparametric Bayesian procedures in such parameter identification problems in PDEs. In particular, inverse regression models for elliptic equations and stochastic diffusion
models are considered.
In Chapter 2, we study the statistical inverse problem of recovering an unknown function from a linear indirect measurement corrupted by additive Gaussian white noise. We employ a nonparametric Bayesian approach with standard Gaussian priors, for which the posterior-based reconstruction corresponds to a Tikhonov regulariser with a reproducing kernel Hilbert space norm penalty. We prove a semiparametric Bernstein–von Mises theorem for a large collection of linear functionals of the unknown, implying that semiparametric posterior estimation and uncertainty quantification are valid and optimal from a frequentist point of view. The general result is applied to three concrete examples that cover both the mildly and severely ill-posed cases: specifically, elliptic inverse problems, an elliptic boundary value problem, and the recovery of the initial condition of the heat equation. For the elliptic boundary value problem, we also obtain a nonparametric version of the theorem that entails the convergence of the posterior distribution to a prior-independent infinite-dimensional Gaussian probability measure with minimal covariance. As a consequence, it follows that the Tikhonov regulariser is an efficient estimator, and we derive frequentist guarantees for certain credible balls centred around it.
Chapter 3 is concerned with statistical nonlinear inverse problems. We focus on the prototypical example of recovering the unknown conductivity function in an elliptic PDE in divergence form from discrete noisy point evaluations of the PDE solution. We study the statistical performance of Bayesian nonparametric procedures based on a flexible class of Gaussian (or hierarchical Gaussian) process priors, whose implementation is feasible by MCMC methods. We show that, as the number of measurements increases, the resulting posterior distributions concentrate around the true parameter generating the data, and derive a convergence rate, algebraic in inverse sample size, for the estimation error of the associated posterior means.
Finally, in Chapter 4 we extend the posterior consistency analysis to dynamical models based on stochastic differential equations. We study nonparametric Bayesian models for reversible multi-dimensional diffusions with periodic drift. For continuous observation paths, reversibility is exploited to prove a general posterior contraction rate theorem for the drift gradient vector field under approximation-theoretic conditions on the induced prior for the invariant measure. The general theorem is applied to Gaussian priors and p-exponential priors, which are shown to converge to the truth at the minimax optimal rate over Sobolev smoothness classes in any dimension.
Chapter 1 is dedicated to introducing the statistical models considered in Chapters 2 - 4, and to providing an overview of the theoretical results derived therein. The main theorems of Chapter 2 and Chapter 3 are illustrated via the results of simulations, and detailed comments are provided on the implementation.Richard Nickl’s ERC grant No. 647812; EPSRC grant EP/L016516/1 for the
Cambridge Centre for Analysi
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Computational Inverse Problems
Inverse problem typically deal with the identification of unknown quantities from indirect measurements and appear in many areas in technology, medicine, biology, finance, and econometrics. The computational solution of such problems is a very active, interdisciplinary field with close connections to optimization, control theory, differential equations, asymptotic analysis, statistics, and probability. The focus of this workshop was on hybrid methods, model reduction, regularization in Banach spaces, and statistical approaches
Image Restoration
This book represents a sample of recent contributions of researchers all around the world in the field of image restoration. The book consists of 15 chapters organized in three main sections (Theory, Applications, Interdisciplinarity). Topics cover some different aspects of the theory of image restoration, but this book is also an occasion to highlight some new topics of research related to the emergence of some original imaging devices. From this arise some real challenging problems related to image reconstruction/restoration that open the way to some new fundamental scientific questions closely related with the world we interact with