1 1 2 1 1 2 2 2 1 3 2 3 1 4 2 4 1 2 3 chol V$_{1}$ 1 chol V$_{2}$ 1 chol V$_{2}$ 2 mean 1 mean 2 0.95 1.00 1.05 1.10 0.1 0.2 0.3 0.4 0.5 1.2 1.3 1.4 1.5 0.5 0.6 0.7 −0.01 0.00 0.01 0.02 0.03 −0.050 −0.025 0.000 0.025 0.050 0.075 −0.05 0.00 0.05 0.00 0.02 0.04 0.0 0.1 0.2 0.3 −0.2 −0.1 0.0 33.80 33.85 33.90 0.80 0.84 0.88 −0.01 0.00 0.01 0.02 0.03 0.19 0.20 0.21 0.22 1.01 1.02 1.03 0.005 0.010 0.015 500 550 600 650 700 500 550 600 650 700 500 550 600 650 700 500 550 600 650 700 Iteration (x 1000) value Figure 5: HMC trace plot for value of population-level parameters in hierarchical model example, starting at Iteration 500,000. Every 500th iteration is plotted. about 8.7 seconds. The time to sample 1,000 proposals from an MVN distribution, and compute the MVN densities for each, is about .89 seconds. Therefore, to collect and evaluate 16.5 million proposal samples (to get 300 posterior draws) would take about 44 hours. But this is for a single processing core. Using all 12 cores on our Mac Pro, the sampling time is reduced to 220 minutes. We discuss the scalability of the component steps of the algorithm in Section 4. Access to more processing nodes could reduce this time even further. It is the ability to collect posterior samples in parallel that gives our method its greatest advantage over MCMC methods. One could run multiple MCMC chains in parallel, but this involves waiting until all of the chains, individually, converge to the target posterior, before we could begin collecting samples for infer- ence. Even then, there is no way to confirm that the chain has, in fact, converged. We can draw inferences about the accuracy of our method by comparing the estimated marginal densities to those that we get from HMC. Note that the HMC estimates are accurate only if all of the chains converge to the target density, and we have a large-enough effective sample size. This condition clearly does not hold, but it is sufficiently close for the majority of the population-level parameters for us to use HMC samples as a baseline standard. Figure 6 is a comparison of the 18