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Audio inpainting is defined as the process of restoring the damaged segments of an audio signal, based on the known signal values and prior information about the signal. In this paper, we formulate the problem in a Bayesian framework and adopt an efficient sparsity inducing Students-t prior distribution, assumed for the discrete cosine transform coefficients, applied on the signal. We also propose a variational Bayesian algorithm for inpainting, that performs approximate, though tractable, inference. Lastly, experiments demonstrate the efficiency of the proposed methodology when used for declipping audio signals, by comparing with the state-of-the-art.