The’sparse Bayesian’modelling approach, as exemplified by the’relevance vector machine’, enables sparse classification and regression functions to be obtained by linearlyweighting a small number of fixed basis functions from a large dictionary of potential candidates. Such a model conveys a number of advantages over the related and very popular’support vector machine’, but the necessary’training’procedure-optimisation of the marginal likelihood function is typically much slower. We describe a new and highly accelerated algorithm which exploits recently-elucidated properties of the marginal likelihood function to enable maximisation via a principled and efficient sequential addition and deletion of candidate basis functions.