Variational Relevance Vector Machines

The Support Vector Machine (SVM) of Vapnik [9] has become widely established as one of the leading approaches to pattern recognition and machine learning. It expresses predictions in terms of a linear combination of kernel functions centred on a subset of the training data, known as support vectors. Despite its widespread success, the SVM suffers from some important limitations, one of the most significant being that it makes point predictions rather than generating predictive distributions. Recently Tipping [8] has formulated the Relevance Vector Machine (RVM), a probabilistic model whose functional form is equivalent to the SVM. It achieves comparable recognition accuracy to the SVM, yet provides a full predictive distribution, and also requires substantially fewer kernel functions. The original treatment of the RVM relied on the use of type II maximum likelihood (the ‘evidence framework’) to provide point estimates of the hyperparameters which govern model sparsity. In this paper we show how the RVM can be formulated and solved within a completely Bayesian paradigm through the use of variational inference, thereby giving a posterior distribution over both parameters and hyperparameters. We demonstrate the practicality and performance of the variational RVM using both synthetic and real world examples. 1 RELEVANCE VECTORS Many problems in machine learning fall under the heading of supervized learning, in which we are given a set of input vectors X = {xn}Nn=1 together with corresponding target values T = {tn} N n=1 . The goal is to use this training data, together with any pertinent prior knowledge, to make predictions of t for new values of x. We can distinguish two distinct cases: regression, in which t is a continuous variable, and classification, in which t belongs to a discrete set. Here we consider models in which the prediction y(x,w) is expressed as a linear combination of basis functions φm(x) of the form