Bernard Silverman, University of Bristol, UK. 

Thresholding and Empirical Bayes:  Finding both needles and hay in haystacks 


Thresholding is a standard nonlinear approach to the processing of empirical 
wavelet coefficients and many other statistical problems where the underlying 
model may, in some sense, be sparse. However, the classical approach to 
thresholding may be very inefficient if it is applied to signals whose 
sparsity is not tuned to the threshold in use.  The notion that a signal may 
or may not be sparse can be expressed in a Bayesian framework by an appropriate 
prior distribution, whereby the parameters each have a mixed distribution, the 
mixture being between an atom of probability at zero and a heavy-tailed 
distribution of some sort.   The mixing weight determines the sparsity of the 
distribution, and the choice corresponds to the choice of mixing weight. An 
empirical Bayes approach to this choice is extremely promising, both in theory 
and in practice. It gives excellent adaptivity, with optimum rates of convergence 
over parameter families corresponding to both dense and sparse signals; and it 
performs very well in practical studies, both in the wavelet case and in the 
case of direct estimation of sparse and dense signals.