We develop a Bayesian "testimation" procedure for recovering a high-dimensional vector observed in the white noise.The components of the unknown vector are first simultaneously tested for significance.The "significant" ones are then estimated from the data, while others are discarded.The resulting Bayesian tcstimator leads to a general hard thrcsholding rule which accommodates many of the known thresholding and model selection procedures as its particular cases.From a frequentist view, such a procedure corresponds to a penalized likelihood estimation with the complexity penalty associated with the chosen prior distribution on the number of non-zero components of the unknown vector.We discuss its optimality in a rather general setting under very mild conditions on the prior and specify the class of priors for which it is adaptively minimax for a wide range of sparse sequences.This is a joint work with Vadim Grinshtein, the Open University of Israel and Marianna Pensky,University of Central Florida.