Bayesian Wavelet Analysis with a Model Complexity Prior

SUMMARY In this paper Bayesian methods for the selection and shrinkage of wavelet coeecients are considered. The wavelet coeecients are assigned a zero mean normal prior with precision matrix. This allows for a more general model than has been previously studied and uncertainty in can be controlled through a hyperprior. It is argued that model interpretability is improved by foresaking the conjugate Wishart prior for in favour of a non-conjugate hyperprior that penalises the degrees of freedom of the model. The degrees of freedom provides a single intuitive measure of model complexity that accommodates both the number of coeecients and the extent of the shrinkage on them. We compare three diierent sampling strategies and comment on their performance. 1. INTRODUCTION Wavelet analysis has quickly established itself as a standard method for the analysis and smoothing of time series. Of particular importance is the application to the denoising and compression of signals (e.g. Bruce and Gao, 1996). In this paper we consider wavelets within a Bayesian framework. The main motivation is to generalise previous work (Clyde, Parmigiani and Vidakovic, 1998; M uller and Vidakovic, 1999) and to make prior assumptions that can be compared to conventional model selection criteria. In addition we compare the relative performance of three well-used sampling strategies and assess their relative merits. Bayesian analysis begins with the elicitation of prior distributions on the wavelet coeecients that reeect subjective beliefs about the probable values of the coeecients. This prior should be interpretable enough to allow us to use it to reeect our prior beliefs on the true signal. The most commonly used prior, and the one we consider here, is a zero mean normaìshrinkage' prior on the vector of wavelet coeecients with some precision matrix. A suitable value of is rarely known a priori and setting it can prove problematic as ultimately controls the complexity, and hence smoothness, of the wavelet reconstruction. We use a hierarchical Bayesian approach which accommodates uncertainty in by including a hyperprior. We refrain from using the conjugate Wishart prior for because this suuers from a lack of interpretability, eeectively diluting uncertainty on into uncertainty on this hyper-hyper prior. Instead we assign a non-conjugate prior distribution over , motivated by the the degrees of freedom of the wavelet network. We choose this as it ooers a natural vehicle for expressing prior knowledge about the underlying function.