  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.
