Dynamic compressed sensing (DCS) has recently gained popularity as a successful approach to recovering dynamic sparse signals. In this paper, we attack the problem from a Bayesian perspective. The proposed model imposes sparse constraints on both the unknown sparse signal and its temporal innovation via t priors. Due to the conjugacy between the priors and likelihoods, we are able to propose a computationally efficient mean-field variational Bayes algorithm to learn the model without parameter tuning. We consider both the online and offline scenarios, and demonstrate via numerical experiments that the proposed methods are superior to alternatives in terms of both reconstruction accuracy and computational time.