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This paper introduces new and practically relevant non-Gaussian priors for the Sparse Bayesian Learning (SBL) framework applied to the Multiple Measurement Vector (MMV) problem. We extend the Gaussian Scale Mixture (GSM) framework to model prior distributions for row vectors, exploring the use of shared and different hyperparameters across different measurements. We propose Expectation Maximization (EM) based algorithms to estimate the parameters of the prior density along with the hyperparameters. To promote sparsity more effectively in a non-Gaussian setting, we show the importance of incorporating learning of the parameters of the mixing density. Such an approach effectively utilizes the common support notion in the MMV problem and promotes sparsity without explicitly imposing a sparsity-promoting prior, indicating the methods’ robustness to model mismatches. Numerical simulations are provided to compare the proposed approaches with the existing SBL algorithm for the MMV problem.