We propose a family of multi-task learning algorithms for collaborative computer aided diagnosis which aims to diagnose multiple clinically-related abnormal structures from medical images. Our formulations eliminate features irrelevant to all tasks, and identify discriminative features for each of the tasks. A probabilistic model is derived to justify the proposed learning formulations. By equivalence proof, some existing regularization-based methods can also be interpreted by our probabilistic model as imposing a Wishart hyperprior. Convergence analysis highlights the conditions under which the formulations achieve convexity and global convergence. Two real-world medical problems: lung cancer prognosis and heart wall motion analysis, are used to validate the proposed algorithms.