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The Horn-Schunck (HS) optical flow method is widely employed to initialize many motion estimation algorithms. In this work, a variational Bayesian approach of the HS method is presented where the motion vectors are considered to be spatially varying Student's t-distributed unobserved random variables and the only observation available is the temporal image difference. The proposed model takes into account the residual resulting from the linearization of the brightness constancy constraint by Taylor series approximation, which is also assumed to be a spatially varying Student's t-distributed observation noise. To infer the model variables and parameters we recur to variational inference methodology leading to an expectation-maximization (EM) framework in a principled probabilistic framework where all of the model parameters are estimated automatically from the data.