ARX models in which the innovations are described by as Studentís t distributed instead of the typical Gaussian noise has certain advantages. In this paper, we consider such a model in a Bayesian setting and develop numerical procedures based on Markov Chain Monte Carlo methods to perform inference. This model includes automatic order determination by two alternative methods, based on a parametric model order and a sparseness prior, respectively. Our choice of distribution of the innovations provides an increased robustness to data anomalies, such as outliers and missing observations. This is illustrated in three numerical studies using both simulated data and real EEG data.