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• A DOA estimation method based on variational Bayesian inference is proposed. • The minimum hole array is adopted to maximize the array aperture based sparse representation. • A sparsity-inducing Bayesian framework with three-stage hierarchical Laplace priors is developed. • The unknown noise is eliminated by a selection matrix. • A refinement operation based on Newton's method is developed to compensate the off-grid errors. For direction-of-arrival (DOA) estimation problems, sparse Bayesian learning (SBL) has achieved excellent estimation performance, especially in sparse arrays. However, numerous SBL-based methods with hyperparameters assigned to Gaussian priors cannot enhance sparsity well, and mainly focus on the nested array (NA) or the co-prime array (CPA) that cause relatively large degree of freedom (DOF) losses. Based on this, we propose a novel method with a Bayesian framework containing three-stage hierarchical Laplace priors that significantly promote sparsity. Moreover, the proposed method is based on the minimum hole array (MHA) that retains a larger array aperture than NA or CPA after redundancy removal, which is required and achieved simultaneously by a denoising operation. In addition, to correct the intractable off-grid model errors caused by grid mismatch, a new refinement operation is developed. And, the refinement empirically outperforms others based on Taylor expansion. Extensive simulations are presented to confirm the superiority of the proposed method beyond state-of-the-art methods. [ABSTRACT FROM AUTHOR]