This paper introduces a general Bayesian framework for obtaining sparse solutions to re- gression and classi cation tasks utilising models linear in the parameters. Although this framework is fully general, we illustrate our approach with a particular specialisation that we denote the `relevance vector machine' (RVM), a model of identical functional form to the popular and state-of-the-art `support vector machine' (SVM). We demonstrate that by exploiting a probabilistic Bayesian learning framework, we can derive accurate prediction models which typically utilise dramatically fewer basis functions than a comparable SVM while o ering a number of additional advantages. These include the bene ts of probabilis- tic predictions, automatic estimation of `nuisance' parameters, and the facility to utilise arbitrary basis functions (e.g. non-`Mercer' kernels). We detail the Bayesian framework and associated learning algorithm for the RVM, and give some illustrative examples of its application along with some comparative benchmarks. We o er some explanation for the exceptional degree of sparsity obtained, and discuss and demonstrate some of the advantageous features, and potential extensions, of Bayesian relevance learning.
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