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
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Abstract
Weinberg (2012) described a constructive algorithm for computing the marginal likelihood,
Z, from a Markov chain simulation of the posterior distribution. Its key point is:
the choice of an integration subdomain that eliminates subvolumes with poor sampling
owing to low tail-values of posterior probability. Conversely, this same idea may
be used to choose the subdomain that optimizes the accuracy of Z. Here, we explore
using the simulated distribution to define a small region of high posterior probability,
followed by a numerical integration of the sample in the selected region using the
volume tessellation algorithm described in Weinberg (2012). Even more promising is
the resampling of this small region followed by a naive Monte Carlo integration. The
new enhanced algorithm is computationally trivial and leads to a dramatic improvement
in accuracy. For example, this application of the new algorithm to a four-component
mixture with random locations in 16 dimensions yields accurate evaluation of Z with
5% errors. This enables Bayes-factor model selection for real-world problems that
have been infeasible with previous methods.