  Markov chain Monte Carlo (MCMC) methods require a large number of samples to
approximate a posterior distribution, which can be costly when the likelihood
or prior is expensive to evaluate. The number of samples can be reduced if we
can avoid repeated samples and those that are close to each other. This is the
idea behind deterministic sampling methods such as Quasi-Monte Carlo (QMC).
However, the existing QMC methods aim at sampling from a uniform hypercube,
which can miss the high probability regions of the posterior distribution and
thus the approximation can be poor. Minimum energy design (MED) is a recently
proposed deterministic sampling method, which makes use of the posterior
evaluations to obtain a weighted space-filling design in the region of
interest. However, the existing implementation of MED is inefficient because it
requires several global optimizations and thus numerous evaluations of the
posterior. In this article, we develop an efficient algorithm that can generate
MED with few posterior evaluations. We also make several improvements to the
MED criterion to make it perform better in high dimensions. The advantages of
MED over MCMC and QMC are illustrated using an example of calibrating a
friction drilling process.
