Bayesian inference with rescaled Gaussian process priors

Vaart, Aad van der; Zanten, Harry van

doi:10.1214/07-ejs098

Cited by 58 publications

(72 citation statements)

References 27 publications

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“…() among others. Posterior asymptotic properties, which are primarily driven by the structure of their reproducing kernel Hilbert space, were studied by Tokdar & Ghosh (), Ghosal & Roy (), Choi & Schervish (), van der Vaart and van Zanten (), Castillo () & Castillo (), Castillo et al . () and Bhattacharya et al .…”

Section: Introductionmentioning

confidence: 99%

“…Thus, a random series prior may be regarded as a flexible alternative to a Gaussian process prior. It is interesting to note that the theory of posterior contraction for Gaussian process priors established in van der Vaart & van Zanten (), van der Vaart & van Zanten (), & van der Vaart & van Zanten () use deep properties of Gaussian processes, while relatively elementary techniques lead to comparable posterior contraction rates for finite random series priors. Posterior computation for Gaussian process priors often need reversible jump MCMC procedures (Tokdar, ) typically with a large number of knots to approximate a given Gaussian process.…”

Section: Introductionmentioning

confidence: 99%

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Adaptive Bayesian Procedures Using Random Series Priors

Shen

Ghosal

2015

Scandinavian J Statistics

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We consider a general class of prior distributions for nonparametric Bayesian estimation which uses finite random series with a random number of terms. A prior is constructed through distributions on the number of basis functions and the associated coefficients. We derive a general result on adaptive posterior contraction rates for all smoothness levels of the target function in the true model by constructing an appropriate ‘sieve’ and applying the general theory of posterior contraction rates. We apply this general result on several statistical problems such as density estimation, various nonparametric regressions, classification, spectral density estimation and functional regression. The prior can be viewed as an alternative to the commonly used Gaussian process prior, but properties of the posterior distribution can be analysed by relatively simpler techniques. An interesting approximation property of B‐spline basis expansion established in this paper allows a canonical choice of prior on coefficients in a random series and allows a simple computational approach without using Markov chain Monte Carlo 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 Tecator data using functional regression models.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Adaptive Bayesian Procedures Using Random Series Priors

Shen

Ghosal

2015

Scandinavian J Statistics

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“…We included the SE estimator as it is commonly used, and, if the scaling parameter λ is suitably chosen, it has optimal asymptotic convergence rate for all functions in F (A. van der Vaart & van Zanten, 2007). As mentioned above, in the present case, the regularizer of f is its posterior mean under a Brownian motion prior.…”

Section: Simulation Studymentioning

confidence: 99%

Regression with I-priors

Bergsma

2020

Econometrics and Statistics

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We consider the problem of estimating the regression function f in the regression model y i = f (x i ) + ε i , where f is assumed to lie in a reproducing kernel Hilbert space (RKHS) and the errors are multivariate normal. This model has wide ranging applications, from regression with a functional covariate to (naive) classification.The main contribution of this paper is a proposal for an objective prior for the regression function f , defined as the distribution maximizing entropy subject to a suitable constraint based on the Fisher information on the regression function. The prior, which we call I-prior, is Gaussian with covariance kernel proportional to the Fisher information, and mean chosen a priori (e.g., 0). The I-prior has the intuitively appealing property that the more information is available about a linear functional of the regression function, the larger its prior variance, and, broadly speaking, the less influential the prior is on the posterior.The I-prior methodology can be used as a principled alternative to Tikhonov regularization, which suffers from well-known theoretical problems which we briefly review.We describe in some detail the case that the regression function lies in the multidimensional fractional Brownian motion RKHS, when the I-prior methodology has some particular appeal. Analysis of some real data sets and a small-scale simulation study show competitive performance of the I-prior methodology, which is implemented in the R-package iprior (Jamil, 2017).

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“…One can taste a variety of recent applications from the books and surveys such as Mandjes [135] and Willinger et al [183] (models of communication networks) Rasmussen and Williams [150] (Machine Learning), van der Vaart and van Zanten [177] (prior models in Bayesian Statistics).…”

Section: Invitation To Further Readingmentioning

confidence: 99%

Lectures on Gaussian Processes

Lifshits¹

2012

SpringerBriefs in Mathematics

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Bayesian inference with rescaled Gaussian process priors

Cited by 58 publications

References 27 publications

Adaptive Bayesian Procedures Using Random Series Priors

Adaptive Bayesian Procedures Using Random Series Priors

Regression with I-priors

Lectures on Gaussian Processes

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