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An empirical Bayes approach to the estimation of possibly sparse sequencesobserved in Gaussian white noise is set out and investigated. The priorconsidered is a mixture of an atom of probability at zero and a heavy-taileddensity \gamma, with the mixing weight chosen by marginal maximum likelihood,in the hope of adapting between sparse and dense sequences. If estimation isthen carried out using the posterior median, this is a random thresholdingprocedure. Other thresholding rules employing the same threshold can also beused. Probability bounds on the threshold chosen by the marginal maximumlikelihood approach lead to overall risk bounds over classes of signalsequences of length n, allowing for sparsity of various kinds and degrees.  The signal classes considered are ``nearly black'' sequences where only aproportion \eta is allowed to be nonzero, and sequences with normalized \ell_pnorm bounded by \eta, for \eta >0 and 0<p\le 2. Estimation error is measured bymean qth power loss, for 0<q\le 2. For all the classes considered, and for allq in (0,2], the method achieves the optimal estimation rate as n\to \infty and\eta \to 0 at various rates, and in this sense adapts automatically to thesparseness or otherwise of the underlying signal. In addition the risk isuniformly bounded over all signals. If the posterior mean is used as theestimator, the results still hold for q>1. Simulations show excellentperformance.

Needles and straw in haystacks: Empirical Bayes estimates of possibly sparse sequences