Sparse Bayesian learning has emerged as a powerful tool to tackle various image classification tasks. The existing sparse Bayesian models usually use independent Gaussian distribution as the prior knowledge for the noise. However, this assumption often contradicts to the practical observations in which the noise is long tail and pixels containing noise are spatially correlated. To handle the practical noise, this paper proposes to partition the noise image into several 2-D groups and adopt the long-tail distribution, i.e., the scale mixture of the matrix Gaussian distribution, to model each group to capture the intragroup correlation of the noise. Under the nonparametric Bayesian estimation, the low-rank-induced prior and the matrix Gamma distribution prior are imposed on the covariance matrix of each group, respectively, to induce two Bayesian correlated group regression (BCGR) methods. Moreover, the proposed methods are extended to the case with unknown group structure. Our BCGR method provides an effective way to automatically fit the noise distribution and integrates the long-tail attribute and structure information of the practical noise into model. Therefore, the estimated coefficients are better for reconstructing the desired data. We apply BCGR to address image classification task and utilize the learned covariance matrices to construct a grouped Mahalanobis distance to measure the reconstruction residual of each class in the design of a classifier. Experimental results demonstrate the effectiveness of our new BCGR model.