  We consider a problem of recovering a high-dimensional vector $\mu$ observed
in white noise, where the unknown vector $\mu$ is assumed to be sparse. The
objective of the paper is to develop a Bayesian formalism which gives rise to a
family of $l_0$-type penalties. The penalties are associated with various
choices of the prior distributions $\pi_n(\cdot)$ on the number of nonzero
entries of $\mu$ and, hence, are easy to interpret. The resulting Bayesian
estimators lead to a general thresholding rule which accommodates many of the
known thresholding and model selection procedures as particular cases
corresponding to specific choices of $\pi_n(\cdot)$. Furthermore, they achieve
optimality in a rather general setting under very mild conditions on the prior.
We also specify the class of priors $\pi_n(\cdot)$ for which the resulting
estimator is adaptively optimal (in the minimax sense) for a wide range of
sparse sequences and consider several examples of such priors.
