Predictive Density Estimation for Multiple Regression

George, Edward I.; Xu, Xinyi

doi:10.1017/s0266466608080213

Cited by 12 publications

(16 citation statements)

References 10 publications

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“…We note that [4] developed minimaxity and dominance conditions on prior distributions for the predictive regression problem with changeable covariance independently of our study. They also proposed a shrinkage prediction where only a subset of the regression coefficients are close to zero.…”

Section: Conclusion and Discussionmentioning

confidence: 99%

Bayesian shrinkage prediction for the regression problem

Kobayashi

Komaki

2008

Journal of Multivariate Analysis

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We consider Bayesian shrinkage predictions for the Normal regression problem under the frequentist Kullback-Leibler risk function.Firstly, we consider the multivariate Normal model with an unknown mean and a known covariance. While the unknown mean is fixed, the covariance of future samples can be different from that of training samples. We show that the Bayesian predictive distribution based on the uniform prior is dominated by that based on a class of priors if the prior distributions for the covariance and future covariance matrices are rotation invariant.Then, we consider a class of priors for the mean parameters depending on the future covariance matrix. With such a prior, we can construct a Bayesian predictive distribution dominating that based on the uniform prior.Lastly, applying this result to the prediction of response variables in the Normal linear regression model, we show that there exists a Bayesian predictive distribution dominating that based on the uniform prior. Minimaxity of these Bayesian predictions follows from these results.

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Section: Conclusion and Discussionmentioning

confidence: 99%

Bayesian shrinkage prediction for the regression problem

Kobayashi

Komaki

2008

Journal of Multivariate Analysis

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“…However, these two covariance structures are generally different when we consider the regression problem (Kobayashi & Komaki, 2008;George & Xu, 2008). We extend the results for this general situation to matrix-variate case.…”

Section: Singular Value Shrinkage Priors Depending On the Future Covamentioning

confidence: 57%

“…George et al (2006) generalized this result and proved that Bayesian predictive densities based on superharmonic priors dominate those based on the Jeffreys prior under the Kullback-Leibler risk. Next, Kobayashi & Komaki (2008) and George & Xu (2008) considered the cases where Σ is not necessarily proportional to Σ. Bayesian predictive densities based on superharmonic priors dominate those based on the uniform prior under the Kullback-Leibler risk also in this general situation. Kobayashi & Komaki (2008) and George & Xu (2008) applied their results to linear regression problems.…”

Section: Introductionmentioning

confidence: 99%

“…Next, Kobayashi & Komaki (2008) and George & Xu (2008) considered the cases where Σ is not necessarily proportional to Σ. Bayesian predictive densities based on superharmonic priors dominate those based on the uniform prior under the Kullback-Leibler risk also in this general situation. Kobayashi & Komaki (2008) and George & Xu (2008) applied their results to linear regression problems. Now, since matrix-variate Normal distributions are special cases of vectorvariate Normal distributions as in (1), the above results also apply to the matrix-variate Normal distributions.…”

Section: Introductionmentioning

confidence: 99%

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Singular value shrinkage priors for Bayesian prediction

Matsuda

Komaki

2015

Biometrika

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We develop singular value shrinkage priors for the mean matrix parameters in the matrix-variate Normal model with known covariance matrices. Introduced priors are superharmonic and put more weight on matrices with smaller singular values. They are a natural generalization of the Stein prior. Bayes estimators and Bayesian predictive densities based on introduced priors are minimax and dominate those based on the uniform prior in finite samples. The risk reduction is large when the true value of the parameter has small singular values. In particular, introduced priors perform stably well in the low rank case. We apply this result to multivariate linear regression problems by considering priors depending on the future samples. Introduced priors are compatible with reduced-rank regression.

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“…George [17][18][19] showed that such a multiple shrinkage estimators can adaptively shrink toward the point or subspace most favored by the data. George and Xu [21] used the same idea to construct multiple shrinkage Bayesian predictive densities for linear regression models. Another possible extension of this work is to consider different ''linear coefficients'' for the mean and the variance of the empirical Bayes predictive densities.…”

Section: Discussionmentioning

confidence: 99%

Empirical Bayes predictive densities for high-dimensional normal models

Zhou

2011

Journal of Multivariate Analysis

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a b s t r a c tThis paper addresses the problem of estimating the density of a future outcome from a multivariate normal model. We propose a class of empirical Bayes predictive densities and evaluate their performances under the Kullback-Leibler (KL) divergence. We show that these empirical Bayes predictive densities dominate the Bayesian predictive density under the uniform prior and thus are minimax under some general conditions. We also establish the asymptotic optimality of these empirical Bayes predictive densities in infinitedimensional parameter spaces through an oracle inequality.

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Predictive Density Estimation for Multiple Regression

Abstract: Suppose we observe X ∼ N m (Aβ, σ 2 I) and would like to estimate the predictive density

Cited by 12 publications

References 10 publications

Bayesian shrinkage prediction for the regression problem

Bayesian shrinkage prediction for the regression problem

Singular value shrinkage priors for Bayesian prediction

Empirical Bayes predictive densities for high-dimensional normal models

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