Bayesian Methods for Function Estimation

Choudhuri, Nidhan; Ghosal, Subhashis

doi:10.1016/s0169-7161(05)25013-7

Cited by 14 publications

(15 citation statements)

References 117 publications

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“…Thus, compared to frequentist approaches, the Bayesian nonparametric approach is more difficult to implement and hence we do not consider this approach in this presentation. We refer the interested reader to [18] for a review of recent developments on this topic.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Properties of Probability Measure Estimators in a Nonparametric Model

Banks¹,

Catenacci²,

Hu³

2015

SIAM/ASA J. Uncertainty Quantification

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We consider probability measure estimation in a nonparametric model using a leastsquares approach under the Prohorov metric framework. We summarize the computational methods and their convergence results that were developed by our group over the past two decades. New results are presented on the bias and the variance due to the approximation and the pointwise asymptotic normality of the approximated probability measure estimator. We propose use of model selection criterion to balance the bias and the variance, and compare the pointwise confidence band constructed using the asymptotic normality results with that obtained by Monte Carlo simulations.

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Section: Introductionmentioning

confidence: 99%

Asymptotic Properties of Probability Measure Estimators in a Nonparametric Model

Banks¹,

Catenacci²,

Hu³

2015

SIAM/ASA J. Uncertainty Quantification

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“…[BG14]), the estimation either of a spectral density for a stationary time series or of the transition density of some ergodic Markov processes (see e.g. [CGR05]). Recently, [Ver13] has studied the case where the observations are dependent, namely linked through a hidden Markov model.…”

Section: Consistency Rate Of Bayesian Proceduresmentioning

confidence: 99%

On some recent advances on high dimensional Bayesian statistics

et al. 2015

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Abstract. This paper proposes to review some recent developments in Bayesian statistics for high dimensional data. After giving some brief motivations in a short introduction, we describe new advances in the understanding of Bayes posterior computation as well as theoretical contributions in non parametric and high dimensional Bayesian approaches. From an applied point of view, we describe the so-called SQMC particle method to compute posterior Bayesian law, and provide a nonparametric analysis of the widespread ABC method. On the theoretical side, we describe some recent advances in Bayesian consistency for a nonparametric hidden Markov model as well as new PAC-Bayesian results for different models of high dimensional regression.Résumé. Cet article propose une vue d'ensemble de récents développements en statistiques Bayésiennes en grande dimension. Après quelques motivations rappelées en introduction, nous présentons des avancéesà la fois algorithmiques et dans la compréhension théorique de méthodes de calculs d'a posteriori Bayésien. En particulier, nous décrivons l'algorithme particulaire SQMC et proposons un point de vue non-paramétrique de la méthode populaire ABC. Nous revenons ensuiteégalement sur des nouvelles contributions en statistiques bayésiennes non paramétriques et en grandes dimensions. Dans ce contexte, nous décrivons des résultats de consistance bayésienne a posteriori pour des modèles non-paramétriques de Markov cachés ainsi que des résultats PAC-bayésiens pour différents modèles de régression.

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“…Very good tutorials [7], [8], [9], [10] bring out the wide variety of fields covered by the approach. A nonparametric or semiparametric model involve one or more infinite dimensional parameters.…”

Section: Bayesian Nonparametric Density Estimationmentioning

confidence: 99%

“…For both spectra all parameters are taken identical : DP concentration parameter α = 1 and N = 100. We use a canonical PT (see [16], [7]) with M = 10, A = {a m = 6 m : m ≤ M } (see Appendix I and [16]) and Δ = 128. We averaged 20000 draws of the Gibbs sampler after 10000 burn in iterations.…”

Section: Applicationmentioning

confidence: 99%

Nonparametric bayesian inference in nuclear spectrometry

Barat

Dautremer

Montagu

2007

2007 IEEE Nuclear Science Symposium Conference Record

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We address the problem of X/γ-ray spectra estimation in the fields of nuclear physics. Bayesian estimation of experimental backgrounds has been studied in [1] involving splines. Since Dirichlet Processes (DP) sit on discrete measures, they provide an appealing prior for photopeaks. On the other hand, in order to tackle the complexity of experimental backgrounds, we consider a Pólya Tree Mixture (PTM) -with suitable parameters yielding distribution continuity -for which predictive densities exhibit better smoothness properties than a single Pólya Tree. Furthermore, it is easy to introduce some physical Compton line approximation formula (e.g. Klein-Nishina) in the base measure of the Pólya Tree, or some physically driven local modifications of the PTM prior parameters. As backgrounds depend on photopeaks locations, we propose a hierarchical model where the PTM is conditioned on the DP. We use a beta prior for the mixing proportion between the DP and the PTM. Energies are not directly observed due to detection devices noises which introduce a convolution of both discrete and continuous measures by an assumed gaussian kernel whose variance is an unknown linear function of energy. Thus, the proposed semiparametric model for experimental data becomes a hierarchical Pólya TreeDirichlet mixture of normal kernels. Besides, observed energies are binned in an histogram introducing additional quantification noise. The quantities of interest are usually posterior functionals of the DP mixing distribution. This implies an inverse problem which is carried out in the framework of finite stick-breaking representation. Thanks to conjugacy, draws from the posterior DP and PTM are easily obtained. The approach yields to a global peaks/background separation while offering spectrum resolution enhancement. The method is illustrated on experimental HPGe spectra.

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Bayesian Methods for Function Estimation

Cited by 14 publications

References 117 publications

Asymptotic Properties of Probability Measure Estimators in a Nonparametric Model

Asymptotic Properties of Probability Measure Estimators in a Nonparametric Model

On some recent advances on high dimensional Bayesian statistics

Nonparametric bayesian inference in nuclear spectrometry

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