<p>Sparse Bayesian learning (SBL) has become one of the principal foundations of machine learning. This paper focuses on the SBL algorithm for two-dimensional (2-D) direction-of-arrival (DOA) estimation with a three-dimensional (3-D) antenna array structured by two L-shaped arrays. The high complexity of the conventional expectation–maximization (EM)-based SBL approach, which requires matrix inversion, is addressed first, where we consider the application of a computationally efficient message passing technique to the E-step in the SBL paradigm in lieu of matrix inversion. In particular, we formulate the sparse parameter estimation problem at hand as local message passing on a factor graph, which is used to visualize the structure of the probabilistic model drawn from this formalism. We describe how posterior beliefs at each node in the factor graph can be updated by sending messages between nodes. The advantages of the proposed inference scheme include the ability to capture complex temporal dependencies among multiple measurement vectors, the significantly reduced derivation complexity of the estimation scheme, and fewer approximations required in computing marginal probabilities in the distributed factor graph than in existing message passing techniques because of the assumed conditional Gaussian prior on the signal’s coefficients. We then introduce quadratic interpolation for signal direction refinement on the basis of the reconstruction result. Finally, we perform pair matching for all estimated angles by using the spatial relationships of these angles. Simulated examples are provided to demonstrate the close to optimal performance of the derived algorithm.</p>