  This paper explores a class of empirical Bayes methods for level-dependent
threshold selection in wavelet shrinkage. The prior considered for each wavelet
coefficient is a mixture of an atom of probability at zero and a heavy-tailed
density. The mixing weight, or sparsity parameter, for each level of the
transform is chosen by marginal maximum likelihood. If estimation is carried
out using the posterior median, this is a random thresholding procedure; the
estimation can also be carried out using other thresholding rules with the same
threshold. Details of the calculations needed for implementing the procedure
are included. In practice, the estimates are quick to compute and there is
software available. Simulations on the standard model functions show excellent
performance, and applications to data drawn from various fields of application
are used to explore the practical performance of the approach. By using a
general result on the risk of the corresponding marginal maximum likelihood
approach for a single sequence, overall bounds on the risk of the method are
found subject to membership of the unknown function in one of a wide range of
Besov classes, covering also the case of f of bounded variation. The rates
obtained are optimal for any value of the parameter p in (0,\infty],
simultaneously for a wide range of loss functions, each dominating the L_q norm
of the \sigmath derivative, with \sigma\ge0 and 0<q\le2.
