Bayesian Model Comparison by Monte Carlo Chaining

Abstract

The techniques of Bayesian inference have been applied with great success to many problems in neural computing including evaluation of regression functions, determination of error bars on predictions, and the treatment of hyper-parameters. However, the problem of model comparison is a much more challenging one for which current techniques have signicant limitations. In this paper we show how an extended form of Markov chain Monte Carlo, called chaining, is able to provide eective estimates of the relative probabilities of dierent models. We present results from the robot arm problem and compare them with the corresponding results obtained using the standard Gaussian approximation framework. 1 Bayesian Model Comparison In a Bayesian treatment of statistical inference, our state of knowledge of the values of the parameters w in a modelM is described in terms of a probability distribution function. Initially this is chosen to be some prior distribution p(wjM), which can be combined with a likelihood function p(Djw;M) using Bayes ' theorem to give a posterior distribution p(wjD;M) in the form p(wjD;M) = p(Djw;M)p(wjM) p(DjM) (1) where D is the data set. Predictions of the model are obtained by performing integrations weighted by the posterior distribution. The comparison of dierent modelsM i is based on their relative probabilities, which can be expressed, again using Bayes ' theorem, in terms of prior probabilities p(M i to give p(M i jD) p(M j jD) p(DjM