By Peter Müller, Brani Vidakovic
This quantity offers an outline of Bayesian equipment for inference within the wavelet area. The papers during this quantity are divided into six elements: the 1st papers introduce uncomplicated options. Chapters partially II discover various methods to earlier modeling, utilizing autonomous priors. Papers within the half III speak about determination theoretic elements of such past versions. partly IV, a few features of previous modeling utilizing priors that account for dependence are explored. half V considers using 2-dimensional wavelet decomposition in spatial modeling. Chapters partially VI talk about using empirical Bayes estimation in wavelet dependent types. half VII concludes the amount with a dialogue of case stories utilizing wavelet dependent Bayesian methods. The cooperation of all participants within the well timed coaching in their manuscripts is drastically well-known. We determined early on that it was once impor tant to referee and seriously assessment the papers which have been submitted for inclusion during this quantity. For this large activity, we trusted the carrier of diverse referees to whom we're such a lot indebted. we're additionally thankful to John Kimmel and the Springer-Verlag referees for contemplating our suggestion in a truly well timed demeanour. Our distinct thank you visit our spouses, Gautami and Draga, for his or her support.
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Additional info for Bayesian Inference in Wavelet-Based Models
References Benedetto, J. J. and Frazier, M. W. (1994) Wavelets: Mathematics and Applications, CRC Press, Bocan Raton, Florida. Bloomfield, P. (1976) Fourier analysis of time series, an introduction, Wiley, New York. Daubechies, I. (1992) Ten Lectures on Wavelets,. SIAM, Philadelphia. Donoho, D. L. and Johnstone, I. M. (1995). Adapting to unknown smoothness via Wavelet shrinkage, J. Amer. Statist. , 90, 1200-1224. 32 Marron Donoho, D. , Johnstone, I. , Kerkyacharian, G. and Picard, D. (1995). Wavelet shrinkage: Asymptopia?
Note that using this basis, the signal recovery is excellent for both of these data sets. In contrast, the Fourier basis gave excellent recovery of the smooth sinusoidal target, but very poor performance for the step target, while the Haar basis gave excellent recovery of the step target, but poor recovery of the sinusoidal target. The key to understanding this is "signal compression" . The Fourier basis represents the sinusoid with very few coefficients, but needs many for the step. The Haar represents the step with few coefficients, but needs many for the sinusoid.
Overcomplete wavelet dictionaries are obtained by sampling indices more finely. An important example of overcomplete wavelet dictionaries is the non-decimated (or stationary or translation-invariant) wavelet dictionary (see, for example, Coifman & Donoho, 1995; Nason & Silverman, 1995). Atomic decompositions in overcomplete dictionaries are obviously nonunique and one may think about choosing the 'best' possible representation among many available (see, for example, Mallat & Zhang, 1993; Davis, Mallat & Zhang, 1994; Chen, Donoho & Saunders, 1999).