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Traditional bearing estimation techniques perform Nyquist-rate sampling of the received sensor array signals and as a result they require high storage and transmission bandwidth resources. Compressed sensing (CS) theory provides a new paradigm for simultaneously sensing and compressing a signal using a small subset of random incoherent projection coefficients, enabling a potentially significant reduction in the sampling and computation costs. In this paper, we develop a Bayesian CS (BCS) approach for estimating target bearings based on multiple noisy CS measurement vectors, where each vector results by projecting the received source signal on distinct over-complete dictionaries. In addition, the prior belief that the vector of projection coefficients should be sparse is enforced by fitting directly the prior probability distribution with a Gaussian Scale Mixture (GSM) model. The experimental results show that our proposed method, when compared with norm-based constrained optimization CS algorithms, as well as with single-measurement BCS methods, improves the reconstruction performance in terms of the detection error, while resulting in an increased sparsity.