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We investigate whether randomized likelihood (RL) decoding based on sampling techniques can completely replace the compressed sensing (CS) process for sparse recovery. For a Gaussian signal model, we propose a novel iterative Markov chain Monte Carlo (MCMC) sampling-based RL decoding method tailored to the attributes of sparse recovery, termed MCMC-RLD-SR. The proposed iterative MCMC-RLD-SR algorithm incorporates two stages, i.e., rough estimation and fine estimation. The rough estimation is a process of figuring out support candidates for the sparse signal via the Metropolis-Hastings (MH) sampling method, which prevents a nonconvergence issue inherent in the CS problem when applying sampling. The fine estimation is a process of acquiring an estimate of the sparse signal through the Gibbs sampling method based on the support candidates from the rough estimation stage. We prove …