{"title": "Radial Basis Functions: A Bayesian Treatment", "book": "Advances in Neural Information Processing Systems", "page_first": 402, "page_last": 408, "abstract": null, "full_text": "Radial Basis Functions: \na Bayesian treatment \n\nDavid Barber* \n\nBernhard Schottky \n\nNeural Computing Research Group \n\nDepartment of Applied Mathematics and Computer Science \n\nAston University, Birmingham B4 7ET, U.K. \n\nhttp://www.ncrg.aston.ac.uk/ \n\n{D.Barber,B.Schottky}~aston.ac.uk \n\nAbstract \n\nBayesian methods have been successfully applied to regression and \nclassification problems in multi-layer perceptrons. We present a \nnovel application of Bayesian techniques to Radial Basis Function \nnetworks by developing a Gaussian approximation to the posterior \ndistribution which, for fixed basis function widths, is analytic in \nthe parameters. The setting of regularization constants by cross(cid:173)\nvalidation is wasteful as only a single optimal parameter estimate \nis retained. We treat this issue by assigning prior distributions to \nthese constants, which are then adapted in light of the data under \na simple re-estimation formula. \n\n1 \n\nIntroduction \n\nRadial Basis Function networks are popular regression and classification tools[lO]. \nFor fixed basis function centers, RBFs are linear in their parameters and can there(cid:173)\nfore be trained with simple one shot linear algebra techniques[lO]. The use of \nunsupervised techniques to fix the basis function centers is, however, not generally \noptimal since setting the basis function centers using density estimation on the input \ndata alone takes no account of the target values associated with that data. Ideally, \ntherefore, we should include the target values in the training procedure[7, 3, 9]. Un(cid:173)\nfortunately, allowing centers to adapt to the training targets leads to the RBF being \na nonlinear function of its parameters, and training becomes more problematic. \n\nMost methods that perform supervised training of RBF parameters minimize the \n\n\u00b7Present address: SNN, University of Nijmegen, Geert Grooteplein 21, Nijmegen, The \n\nNetherlands. http://wwv.mbfys.kun.nl/snn/ email: davidb