Abstract
Bayesian methods based on hierarchical mixture models have demonstrated excellent mean squared error properties in constructing data dependent shrinkage estimators in wavelets, however, subjective elicitation of the hyperparameters is challenging. In this chapter we use an Empirical Bayes approach to estimate the hyperparameters for each level of the wavelet decomposition, bypassing the usual difficulty of hyperparameter specification in the hierarchical model. The EB approach is computationally competitive with standard methods and offers improved MSE performance over several Bayes and classical estimators in a wide variety of examples.
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References
Abramovich, F., Sapatinas, T., and Silverman, B.W. (1998). “Wavelet Thresholding via a Bayesian Approach,” Journal of the Royal Statistics Society, Series B, 60, 725–749.
Akaike, H. (1973). Information theory and an extension of the maximum likelihood principle. In 2nd International Symposium on Information Theory, Eds B.N. Petrov and F. Csaki, pp. 267–281. Budapest: Akademia Kiado.
Box, G.E.P. and Tiao, G.C. (1973). Bayesian Inference in Statistical Analysis, Wiley, NY.
Chipman, H., Kolaczyk, E., and McCulloch, R. (1997). “Adaptive Bayesian Wavelet Shrinkage”, Journal of the American Statistical Association, 92, 1413–1421.
Clyde, M. and George, E.I. (1998). “Robust Empirical Bayes Estimation in Wavelets”, ISDS Discussion Paper 98-21/ http://www.isds.duke.edu/.
Clyde, M., Parmigiani, G., Vidakovic, B. (1998). “Multiple Shrinkage and Subset Selection in Wavelets,” Biometrika, 85, 391–402.
Dempster, A.P. Laird, N.M. and Rubin, D.B. (1977). Maximum likelihood estimation from incomplete data via the EM algorithm (with discussion). Journal of the Royal Statistical Society, Series B, 39, 1–38.
Donoho, D.L., and Johnstone, I.M., (1994). “Ideal spatial adaptation by wavelet shrinkage,” Biometrika, 81, 256–425.
Donoho, D. and Johnstone, I. (1995). “Adapting to Unknown Smoothness via Wavelet Shrinkage,” Journal of the American Statistical Association, 90, 1200–1224.
Donoho, D., Johnstone, I., Kerkyacharian, G., and Picard, D. (1995). “Wavelet shrinkage: Asymptopia?” Journal of the Royal Statistical Society, Series B, 57, 301–369.
Foster, D. and George, E. (1994). The risk inflation criterion for multiple regression. Annals of Statistics 22, 1947–1975.
George, E.I. (1986). “Minimax multiple shrinkage estimation,” Annals of Statistics, 14, 188–205.
George, E.I. and Foster, D.P. (1997). “Empirical Bayes Variable Selection”, Tech Report, University of Texas at Austin.
George, E.I. and McCulloch, R. (1993). Variable selection via Gibbs sampling. Journal of the American Statistical Association 88, 881–889.
Johnstone, I.M. and Silverman, B.W. (1997). “Wavelet Threshold Estimators for Data with Correlated Noise,” Journal of the Royal Statistical Society, Series B, 59, 319–351.
Johnstone, I.M. and Silverman, B.W. (1998). “Empirical Bayes approaches to mixture problems and wavelet regression”, Technical report. Department of Mathematics, University of Bristol.
Neal, R. M. and Hinton, G. E. (1998) “A view of the EM algorithm that justifies incremental, sparse, and other variants”, in M. I. Jordan (editor) Learning in Graphical Models, Dordrecht: Kluwer Academic Publishers, pages 355–368.
Schwarz, G. (1978). Estimating the dimension of a model. Annals of Statistics 6, 461–464.
Tanner, M.A. (1996). Tools for statistical inference: methods for the exploration of posterior distributions and likelihood functions. New York: Springer, 3rd Edition. Chapter 4, pages 64–89.
Yau, P. and Kohn, R. (1999). “Wavelet Nonparametric Regression Using Basis Averaging”. To appear in Bayesian Inference in Wavelet Based Models eds P. Müller and B. Vidakovic. Springer-Verlag.
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Clyde, M.A., George, E.I. (1999). Empirical Bayes Estimation in Wavelet Nonparametric Regression. In: Müller, P., Vidakovic, B. (eds) Bayesian Inference in Wavelet-Based Models. Lecture Notes in Statistics, vol 141. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-0567-8_19
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DOI: https://doi.org/10.1007/978-1-4612-0567-8_19
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