Abstract
In this chapter, we will provide an overview of the current status of research involving Bayesian inference in wavelet nonparametric problems. In many statistical applications, there is a need for procedures to (i) adapt to data and (ii) use prior information. The interface of wavelets and the Bayesian paradigm provide a natural terrain for both of these goals.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Abramovich, F., Sapatinas, T. and Silverman, B. W. (1998), “Wavelet thresholding via Bayesian approach,” Journal of the Royal Statistical Society, 60, No.3.
Bielza, C., and Vidakovic, B. (1996), “Time-dependent wavelet shrinkage,” Discussion Paper, 96–24, ISDS, Duke University.
Brunk, H. (1978), “Univariate density estimation by orthogonal series,” Biometrika, 65, 3, 521–528.
Carlin, B. and Chib, S. (1995), “Bayesian model choice via Markov chain Monte Carlo,” Journal of the Royal Statistical Society B, 57, 473–484.
Chencov, N. N. (1962), “Evaluation of an unknown distribution density from observations,” Doklady, 3, 1559–1562.
Chipman, H., McCulloch, R., and Kolaczyk, E. (1997), “Adaptive Bayesian Wavelet Shrinkage,” Journal of the American Statistical Association, 92, 1413–1421.
Clyde, M., DeSimone, H., and Parmigiani, G. (1996), “Prediction via orthogonalized model mixing,” Journal of the American Statistical Association, 91, 1197–1208.
Clyde, M., and George, E. (1998), “Empirical wavelet estimation in wavelets,” Discussion Paper, ISDS, Duke University
Clyde, M., Parmigiani, G., and Vidakovic, B. (1998), “Multiple shrinkage and subset selection in wavelets,” Biometrika, to appear.
Crouse, M., Nowak, R., and Baraniuk, R. (1997), “Statistical signal processing using wavelet-domain hidden Markov models,” Proceedings of SPIE, Wavelet Applications in Signal and Image Processing V, vol. 3169, 248–259.
Daubechies, I. (1992), Ten Lectures on Wavelets. S.I.A.M., Philadelphia.
Delyon, B., and Juditsky, A. (1993), “Wavelet estimators, global error measures: Revisited,” Publication interne, 782, IRISA-INRIA, France.
Donoho, D., and Johnstone, I. (1994), “Ideal spatial adaptation by wavelet shrinkage,” Biometrika, 81, 425–455.
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 Pickard, D. (1996), “Density Estimation by Wavelet Thresholding,” The Annals of Statistics. 24, 508–539.
George, E.I., and McCulloch, R. (1997), “Approaches to Bayesian variable selection,” Statistica Sinica, 7, 339–373.
Green, P.J., Sapatinas, T., and Vidakovic, B. (1998), “Fully Bayesian adaptive wavelet shrinkage,” Manuscript in Preparation.
Hall, P., and Patil, P. (1995), “Formulae for the mean integrated square error of non-linear wavelet based density estimators,” The Annals of Statistics, 23, 905–928.
Hernández, E., and Weiss, G. (1996), A First Course on Wavelets, Boca Raton: CRC Press Inc.
Huerta, G. (1997), “Bayes wavelet shrinkage and applications to data denoising,” In electronic proceedings of International Workshop on Wavelets in Statistics, Duke University, 12–13 October 1997.
Huang, H-C., and Cressie, N. (1997), “Deterministic/stochastic wavelet decomposition for recovery of signal from noisy data,” Technical Report, 97–23, Department of Statistics, Iowa State University.
Johnstone, I., and Silverman B. W. (1996), “Wavelet threshold estimators for data with correlated noise,” Journal of the Royal Statistical Society B, 59, 319–351.
Kohn R, and Marron J. S. (1997), “Bayesian wavelet shrinkage,” International Workshop on Wavelets in Statistics, Duke University, 12–13 October 1997.
Lina, J-M., and MacGibbon, B. (1997), “Non-Linear shrinkage estimation with complex Daubechies wavelets,” Proceedings of SPIE, Wavelet Applications in Signal and Image Processing V, vol. 3169, 67–79.
Lu, H. H-S., Huang, S-Y., and Tung Y-C. (1997), “Wavelet shrinkage for nonparametric mixed-effects models,” Technical Report, Institute of Statistics, National Chiao Tung University. Also in: Electronic proceedings of International Workshop on Wavelets in Statistics, Duke University, 12–13 October 1997.
Malfait, M., and Roose, D. (1995), “Wavelets and Markov random fields in a Bayesian framework,” In Wavelets and Statistics (eds: A. Antoniadis and G. Oppenheim), pp. 225–238, Lecture Notes in Statistics, New York: Springer-Verlag
Malfait, M., Jansen, M., and Roose, D. (1996), “Bayesian approach to wavelet-based image processing,” In Proceedings of The Joint Statistical Meetings, Chicago, August 1996.
Mallat, S. (1989), “A theory for multiresolution signal decomposition: The wavelet representation,” IEEE Trans. Pattern Anal. Machine Intell., 11, 674–693.
Meyer, Y. (1992), Wavelets and Operators, Cambridge.
Müller, P., and Vidakovic, B. (1995), “Bayesian inference with wavelets: Density estimation,” Discussion Paper, 95–33, ISDS, Duke University.
Ogden, T. (1996), “Wavelets in Bayesian change-point analysis,” Technical Report, Department of Statistics, University of South Carolina.
Ogden, T. (1997), Essential Wavelets for Statistical Applications and Data Analysis, Birkhäuser, Boston.
Pesquet, J., Krim, H., Leporini, D., and Hamman, E. (1996), “Bayesian approach to best basis selection,” In: IEEE International Conference on Acoustics, Speech, and Signal Processing, 5, 2634–2637.
Rios Insua, D., and Vidakovic, B. (1998), “Wavelet-based random densities,” Computational Statistics, to appear.
Roeder, K. (1990), “Density estimation with confidence sets exemplified by superclusters and voids in the galaxies,” Journal of the American Statistical Association, 85 617–624.
Ruggeri, F., and Vidakovic, B. (1998), “A Bayesian decision theoretic approach to wavelet thresholding,” Statistica Sinica, to appear.
Simoncelli, E., and Adelson, E. (1996), “Noise removal via Bayesian wavelet coring,” Presented at: 3rd IEEE International Conference on Image Processing, Lausanne, Switzerland.
Tierney, L. (1994), “Markov chains for exploring posterior distributions,” The Annals of Statistics, 22, 1701–1728.
Timmermann K. E., and Nowak, R. D.,(1997), “Multiscale Bayesian Estimation of Poisson Intensities,” Conference Record of the Thirty-First Asilomar Conference on Signals, Systems, and Computers, IEEE Computer Society PRESS, Los Alamitos CA.
Vannucci, M. (1996), Sull’Applicazione delle “Wavelets” in Statitstica, Tesi di dottorato, Dipartimento di Statistica G. Parenti, University degli Studi di Firenze.
Vannucci, M., and Corradi, F. (1997), “Some findings on the covariance structure of wavelet coefficients: Theory and models in a Bayesian perspective,” Report UKC/IMS, 97–05, Institute of Maths and Stats, University of Kent at Canterbury.
Vannucci, M., and Vidakovic, B. (1995), “Preventing the Dirac disaster: Wavelet based density estimation,” Discussion Paper, 95–27, ISDS, Duke University.
Vidakovic, B. (1998), “Nonlinear wavelet shrinkage with Bayes rules and Bayes factors,” Journal of the American Statistical Association, 93 173–179.
Vidakovic, B., and Müller, P. (1995), “Wavelet shrinkage with affine Bayes rules with applications,” Discussion Paper, 95–34, ISDS, Duke University.
Wahba, G. (1981), “Data-based optimal smoothing of orthogonal series density estimates,” The Annals of Statistics, 9, 146–156.
Walter, G.G. (1994), Wavelets and Others Orthogonal Systems with Applications, CRC Press, Boca Raton, FL.
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Science+Business Media New York
About this chapter
Cite this chapter
Vidakovic, B. (1998). Wavelet-Based Nonparametric Bayes Methods. In: Dey, D., Müller, P., Sinha, D. (eds) Practical Nonparametric and Semiparametric Bayesian Statistics. Lecture Notes in Statistics, vol 133. Springer, New York, NY. https://doi.org/10.1007/978-1-4612-1732-9_7
Download citation
DOI: https://doi.org/10.1007/978-1-4612-1732-9_7
Publisher Name: Springer, New York, NY
Print ISBN: 978-0-387-98517-6
Online ISBN: 978-1-4612-1732-9
eBook Packages: Springer Book Archive