On posterior distribution of Bayesian wavelet thresholding

Lian, Heng

doi:10.1016/j.jspi.2010.06.016

Cited by 13 publications

(5 citation statements)

References 23 publications

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“…Doing the same calculation as in Example , the resulting rate ϵ n is

n^{- α / (2 α + 1)} {(\log n)}^{α / (2 α + 1) + (1 - t_{2}) / 2}

. This coincides with the adaptation results for white noise models in Lian () and for density estimation and regression models in Rivoirard & Rousseau (). Example Theorem can be used in multi‐dimensional situation as well. Consider the tensor‐product B‐splines (Schumaker, ) as a basis in

{scriptC}^{α} {(0, 1)}^{s}

.…”

Section: General Resultssupporting

confidence: 78%

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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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“…Doing the same calculation as in Example , the resulting rate ϵ n is

n^{- α / (2 α + 1)} {(\log n)}^{α / (2 α + 1) + (1 - t_{2}) / 2}

{scriptC}^{α} {(0, 1)}^{s}

.…”

Section: General Resultssupporting

confidence: 78%

“…(2.22) results for white noise models in Lian (2011) and for density estimation and regression models in Rivoirard & Rousseau (2012b).…”

Section: Example 3 (Polynomial Basis)mentioning

confidence: 99%

Adaptive Bayesian Procedures Using Random Series Priors

Shen

Ghosal

2015

Scandinavian J Statistics

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“…First, we consider a bounded subset H s n (B) = {f ∈ H s n , f H s n < B} of the Sobolev space. The Lemma 1 and 2 of Lian (2011) imply that (30) is positive. Thus, for any B > 0, we have…”

Section: Posterior Consistencymentioning

confidence: 97%

Bayesian Change Point Detection for Functional Data

Ghosal

2018

Preprint

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We propose a Bayesian method to detect change points for functional data. We extract the features of a sequence of functional data by the discrete wavelet transform (DWT), and treat each sequence of feature independently. We believe there is potentially a change in each feature at possibly different time points. The functional data evolves through such changes throughout the sequences of observations. The change point for this sequence of functional data is the cumulative effect of changes in all features. We assign the features with priors which incorporate the characteristic of the wavelet coefficients. Then we compute the posterior distribution of change point for each sequence of feature, and define a matrix where each entry is a measure of similarity between two functional data in this sequence. We compute the ratio of the mean similarity between groups and within groups for all possible partitions, and the change point is where the ratio reaches the minimum. We demonstrate this method using a dataset on climate change.

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“…Several bayesian shrinkage procedures have been proposed in the last years in many statistical fields. Some of them are found in Lian (2011), Beenamol et al (2012), Karagiannis et al (2015), Griffin & Brown (2017) and Torkamani & Sadeghzadeh (2017). Further, priors models in the wavelet domain were proposed since 1990s, as for example a mixture of gaussian distributions by Chipman et al (1997), mixtures of a point mass function at zero and a symmetric distribution were considered by Abramovich et al (1998), Vidakovic (1998) with the use of the t-distribution as the symmetric density in the mixture, Vidakovic & Ruggeri (2001) with the double exponential distribution, Angelini & Vidakovic (2004) with a Γ-Minimax shrinkage rule based on the Bickel distribution, Weibull prior were proposed by Reményi & Vidakovic (2015), Dirichlet-Laplace priors by Bhattacharya et al (2015), the logistic prior by Sousa et al (2021) and the symmetric beta distribution by Sousa et al (2021), among others.…”

Section: Introductionmentioning

confidence: 99%

Asymmetric Prior in Wavelet Shrinkage

Sousa

2022

Rev. colomb. estad.

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In bayesian wavelet shrinkage, the already proposed priors to wavelet coefficients are assumed to be symmetric around zero. Although this assumption is reasonable in many applications, it is not general. The present paper proposes the use of an asymmetric shrinkage rule based on the discrete mixture of a point mass function at zero and an asymmetric beta distribution as prior to the wavelet coefficients in a non-parametric regression model. Statistical properties such as bias, variance, classical and bayesian risks of the associated asymmetric rule are provided and performances of the proposed rule are obtained in simulation studies involving artificial asymmetric distributed coefficients and the Donoho-Johnstone test functions. Application in a seismic real dataset is also analyzed.

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On posterior distribution of Bayesian wavelet thresholding

Cited by 13 publications

References 23 publications

Adaptive Bayesian Procedures Using Random Series Priors

Adaptive Bayesian Procedures Using Random Series Priors

Bayesian Change Point Detection for Functional Data

Asymmetric Prior in Wavelet Shrinkage

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