Results 1  10
of
17
Convergence rates of posterior distributions
 Ann. Statist
, 2000
"... We consider the asymptotic behavior of posterior distributions and Bayes estimators for infinitedimensional statistical models. We give general results on the rate of convergence of the posterior measure. These are applied to several examples, including priors on finite sieves, logspline models, D ..."
Abstract

Cited by 110 (19 self)
 Add to MetaCart
(Show Context)
We consider the asymptotic behavior of posterior distributions and Bayes estimators for infinitedimensional statistical models. We give general results on the rate of convergence of the posterior measure. These are applied to several examples, including priors on finite sieves, logspline models, Dirichlet processes and interval censoring. 1. Introduction. Suppose
Convergence rates of posterior distributions for noniid observations
 Ann. Statist
, 2007
"... We consider the asymptotic behavior of posterior distributions and Bayes estimators based on observations which are required to be neither independent nor identically distributed. We give general results on the rate of convergence of the posterior measure relative to distances derived from a testing ..."
Abstract

Cited by 57 (6 self)
 Add to MetaCart
(Show Context)
We consider the asymptotic behavior of posterior distributions and Bayes estimators based on observations which are required to be neither independent nor identically distributed. We give general results on the rate of convergence of the posterior measure relative to distances derived from a testing criterion. We then specialize our results to independent, nonidentically distributed observations, Markov processes, stationary Gaussian time series and the white noise model. We apply our general results to several examples of infinitedimensional statistical models including nonparametric regression with normal errors, binary regression, Poisson regression, an interval censoring model, Whittle estimation of the spectral density of a time series and a nonlinear autoregressive model.: θ ∈ Θ) be a sequence of statistical experiments with observations X (n), where the parameter set Θ is arbitrary and n is an indexing parameter, usually the sample size. We put a prior distribution Πn on θ ∈ Θ and study the rate of convergence of the posterior
Adaptive Bayesian estimation using a Gaussian random field with inverse Gamma bandwidth
 THE ANNALS OF STATISTICS
, 2009
"... We consider nonparametric Bayesian estimation inference using a rescaled smooth Gaussian field as a prior for a multidimensional function. The rescaling is achieved using a Gamma variable and the procedure can be viewed as choosing an inverse Gamma bandwidth. The procedure is studied from a frequent ..."
Abstract

Cited by 32 (5 self)
 Add to MetaCart
We consider nonparametric Bayesian estimation inference using a rescaled smooth Gaussian field as a prior for a multidimensional function. The rescaling is achieved using a Gamma variable and the procedure can be viewed as choosing an inverse Gamma bandwidth. The procedure is studied from a frequentist perspective in three statistical settings involving replicated observations (density estimation, regression and classification). We prove that the resulting posterior distribution shrinks to the distribution that generates the data at a speed which is minimaxoptimal up to a logarithmic factor, whatever the regularity level of the datagenerating distribution. Thus the hierachical Bayesian procedure, with a fixed prior, is shown to be fully adaptive.
Adaptive Bayesian density estimation with locationscale mixtures
 Electron. J. Statist
"... Abstract: We study convergence rates of Bayesian density estimators based on finite locationscale mixtures of exponential power distributions. We construct approximations of βHölder densities be continuous mixtures of exponential power distributions, leading to approximations of the βHölder dens ..."
Abstract

Cited by 20 (5 self)
 Add to MetaCart
(Show Context)
Abstract: We study convergence rates of Bayesian density estimators based on finite locationscale mixtures of exponential power distributions. We construct approximations of βHölder densities be continuous mixtures of exponential power distributions, leading to approximations of the βHölder 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.
Empirical Bayesian test of the smoothness.
 Math. Methods Statist.
, 2008
"... In the context of adaptive nonparametric curve estimation problem, a common assumption is that a function (signal) to estimate belongs to a nested family of functional classes, parameterized by a quantity which often has a meaning of smoothness amount. It has already been realized by many that the ..."
Abstract

Cited by 3 (0 self)
 Add to MetaCart
In the context of adaptive nonparametric curve estimation problem, a common assumption is that a function (signal) to estimate belongs to a nested family of functional classes, parameterized by a quantity which often has a meaning of smoothness amount. It has already been realized by many that the problem of estimating the smoothness is not sensible. What then can be inferred about the smoothness? The paper attempts to answer this question. We consider the implications of our results to hypothesis testing. We also relate them to the problem of adaptive estimation. The test statistic is based on the marginalized maximum likelihood estimator of the smoothness for an appropriate prior distribution on the unknown signal.
Adaptive Bayesian procedures using random series prior
, 2013
"... We consider a prior for nonparametric Bayesian estimation which uses finite random series with a random number of terms. The prior is constructed through distributions on the number of basis functions and the associated coefficients. We derive a general result on adaptive posterior convergence rates ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
We consider a prior for nonparametric Bayesian estimation which uses finite random series with a random number of terms. The prior is constructed through distributions on the number of basis functions and the associated coefficients. We derive a general result on adaptive posterior convergence rates for all smoothness levels of the function in the true model by constructing an appropriate “sieve ” and applying the general theory of posterior convergence rates. We apply this general result on several statistical problems such as signal processing, density estimation, various nonparametric regressions, classification, spectral density estimation, functional regression etc. The prior can be viewed as an alternative to the commonly used Gaussian process prior, but properties of the posterior distribution can be analyzed by relatively simpler techniques and in many cases allows a simpler approach to computation without using Markov chain MonteCarlo (MCMC) methods. A simulation study is conducted to show that the accuracy of the Bayesian estimators based on the random series prior and the Gaussian process prior are comparable. We apply the method on two interesting data sets on functional regression.
On adaptive Bayesian inference
 Electronic J. Statist
, 2008
"... Abstract: We study the rate of Bayesian consistency for hierarchical priors consisting of prior weights on a model index set and a prior on a density model for each choice of model index. Ghosal, Lember and Van der Vaart [2] have obtained general inprobability theorems on the rate of convergence of ..."
Abstract

Cited by 2 (2 self)
 Add to MetaCart
(Show Context)
Abstract: We study the rate of Bayesian consistency for hierarchical priors consisting of prior weights on a model index set and a prior on a density model for each choice of model index. Ghosal, Lember and Van der Vaart [2] have obtained general inprobability theorems on the rate of convergence of the resulting posterior distributions. We extend their results to almost sure assertions. As an application we study log spline densities with a finite number of models and obtain that the Bayes procedure achieves the optimal minimax rate n −γ/(2γ+1) of convergence if the true density of the observations belongs to the Hölder space C γ [0,1]. This strengthens a result in [1; 2]. We also study consistency of posterior distributions of the model index and give conditions ensuring that the posterior distributions concentrate their masses near the index of the best model.
ANISOTROPIC FUNCTION ESTIMATION USING MULTIBANDWIDTH GAUSSIAN PROCESSES
 SUBMITTED TO THE ANNALS OF STATISTICS
"... In nonparametric regression problems involving multiple predictors, there is typically interest in estimating an anisotropic multivariate regression surface in the important predictors while discarding the unimportant ones. Our focus is on defining a Bayesian procedure that leads to the minimax opti ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
In nonparametric regression problems involving multiple predictors, there is typically interest in estimating an anisotropic multivariate regression surface in the important predictors while discarding the unimportant ones. Our focus is on defining a Bayesian procedure that leads to the minimax optimal rate of posterior contraction (up to a log factor) adapting to the unknown dimension and anisotropic smoothness of the true surface. We propose such an approach based on a Gaussian process prior with dimensionspecific scalings, which are assigned carefullychosen hyperpriors. We additionally show that using a homogenous Gaussian process with a single bandwidth leads to a suboptimal rate in anisotropic cases.