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We consider a problem of recovering a high-dimensional vector $\mu$ observedin white noise, where the unknown vector $\mu$ is assumed to be sparse. Theobjective of the paper is to develop a Bayesian formalism which gives rise to afamily of $l_0$-type penalties. The penalties are associated with variouschoices of the prior distributions $\pi_n(\cdot)$ on the number of nonzeroentries of $\mu$ and, hence, are easy to interpret. The resulting Bayesianestimators lead to a general thresholding rule which accommodates many of theknown thresholding and model selection procedures as particular casescorresponding to specific choices of $\pi_n(\cdot)$. Furthermore, they achieveoptimality 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 resultingestimator is adaptively optimal (in the minimax sense) for a wide range ofsparse sequences and consider several examples of such priors.

On optimality of Bayesian testimation in the normal means problem