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Parsimonious Function Representation and Optimal Predictive Model Selection

This paper proposes an intuitively appealing approach to function approximation that yields both parsimonious functional representations and optimal predictive models. Along the lines of the median probability model, the concept of prevalence is introduced and defined in terms of the posterior model probabilities. The posterior distribution of model size is used as the main device to determine the most probable number of basis elements. The model of size equal to the optimal number of basis functions is formed using the most prevalent basis elements. The resulting model, called prevalence model, is shown to be optimal predictive. The median probability model turns out to be a specific instance the more general and more adaptive prevalence model. The proposed method uses the sample path obtained from the simulation of a birth-and-death process to estimate the prevalence of the basis elements appearing in the expansion of the approximating function. The most probable number of basis elements is also estimated from the same sample path. It is interesting to note that despite using simple non-sparsity inducing priors, the method yields representations that are substantially more parsimonious than the ones obtained from methods based on sparsity-inducing priors. The framework is fairly general, and naturally lends itself to both traditional basis function expansion and the now popular kernel expansion. Several examples are provided to illustrate the performance of the scheme, along with comparisons to existing methods such as the support vector machine and the relevance vector machine.