We propose a novel greedy algorithm for the support recovery of a sparse signal from a small number of noisy measurements. In the proposed method, a new support index is identified for each iteration based on bit-wise maximum a posteriori (B-MAP) detection. This is optimal in the sense of detecting one of the remaining support indices, provided that all the detected indices in the previous iterations are correct. Despite its optimality, it requires an expensive complexity for computing the maximization metric (i.e., a posteriori probability of each remaining support) due to the marginalization of high-dimensional sparse vector. We address this problem by presenting a good proxy (named B-MAP proxy) on the maximization metric which is accurate enough to find the maximum index, rather than an exact probability, Moreover, it is easily evaluated only using vector correlations as in orthogonal matching pursuit (OMP), but the use completely different proxy matrices for maximization. We demonstrate that the proposed B-MAP detection provides a significant gain compared with the existing methods as OMP and MAP-OMP, having the same complexity. Subsequently, we construct the advanced greedy algorithms, based on B-MAP proxy, by leveraging the idea of compressive sampling matching pursuit (CoSaMP) and subspace pursuit (SP). Via simulations, we show that the proposed method outperforms also OMP and MAP-OMP under the frameworks of the advanced greedy algorithms.