A remarkably simple and accurate method for computing the Bayes Factor from a Markov chain Monte Carlo Simulation of the Posterior Distribution in high dimension

摘要

Weinberg (2012) described a constructive algorithm for computing the marginallikelihood, Z, from a Markov chain simulation of the posterior distribution.Its key point is: the choice of an integration subdomain that eliminatessubvolumes with poor sampling owing to low tail-values of posteriorprobability. Conversely, this same idea may be used to choose the subdomainthat optimizes the accuracy of Z. Here, we explore using the simulateddistribution to define a small region of high posterior probability, followedby a numerical integration of the sample in the selected region using thevolume tessellation algorithm described in Weinberg (2012). Even more promisingis the resampling of this small region followed by a naive Monte Carlointegration. The new enhanced algorithm is computationally trivial and leads toa dramatic improvement in accuracy. For example, this application of the newalgorithm to a four-component mixture with random locations in 16 dimensionsyields accurate evaluation of Z with 5% errors. This enables Bayes-factor modelselection for real-world problems that have been infeasible with previousmethods.