Multi-task learning involves solving multiple related learning problems by sharing some common structure for improved generalization performance. A promising idea to multi-task learning is joint feature selection where a sparsity pattern is shared across task specific feature representations. In this paper, we propose a novel Gaussian Process (GP) approach to multi-task learning based on joint feature selection. The novelty of the proposed approach is that it captures the task similarity by sharing a sparsity pattern over the kernel hyper-parameters associated with each task. This is achieved by considering a hierarchical model which imposes a multi-Laplacian prior over the kernel hyper-parameters. This leads to a flexible GP model which can handle a wide range of multi-task learning problems and can identify features relevant across all the tasks. The hyper-parameter estimation results in an optimization problem which is solved using a block co-ordinate descent algorithm. Experimental results on synthetic and real world multi-task learning data sets demonstrate that the flexibility of the proposed model is useful in getting better generalization performance. © 2014 Springer-Verlag.