We propose a novel method called the Relevance Subject Machine (RSM) to solve the person re-identification (re-id) problem. RSM falls under the category of Bayesian sparse recovery algorithms and uses the sparse representation of the input video under a pre-defined dictionary to identify the subject in the video. Our approach focuses on the multi-shot re-id problem, which is the prevalent problem in many video analytics applications. RSM captures the essence of the multi-shot re-id problem by constraining the support of the sparse codes for each input video frame to be the same. Our proposed approach is also robust enough to deal with time varying outliers and occlusions by introducing a sparse, non-stationary noise term in the model error. We provide a novel Variational Bayesian based inference procedure along with an intuitive interpretation of the proposed update rules. We evaluate our approach over several commonly used re-id datasets and show superior performance over current state-of-the-art algorithms. Specifically, for ILIDS-VID, a recent large scale re-id dataset, RSM shows significant improvement over all published approaches, achieving an 11.5% (absolute) improvement in rank 1 accuracy over the closest competing algorithm considered.