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we set the expected path length to 16. 2 By implementing HMC ourselves, we can use the same computer code to compute the log posterior, and its gradient, that we use with our method. This allows the two methods to compete on a level playing field. 

We ran four independent HMC chains for 700,000 iterations each, during a period of more than three weeks. Searching for the posterior mode is considered, in general, to be “good practice” for Bayesian inference, and especially with MCMC; see Step 1 of the “Recommended Strategy for Posterior Simulation” in Section 11.10 of Gelman et al. (2003). Finding the mode of the log poste- rior is the first step of our method anyway, so we initialized one chain there, and the other three at randomly-selected starting values. Figure 4 is a trace plots of the log posterior density. The chains begin to approach each other only after about 500,000 iterations. The panels in Figure 5, are trace plots of the population-level parameters. Some parameter chains appear to have con- verged to each other, with little autocorrelation, but others seem to make no progress at all. Table 1, summarizes the effective sample sizes for estimates of the marginal posterior distributions of population-level parameters, using the final 100,000 samples of the HMC chains. Many of these parameters may require more than a million additional iterations to achieve an effective sample size large enough to make reasonable inferences. 

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The convergence problems are even worse when we consider that each of the 16 steps in the path length for iteration requires one evaluation for both the log posterior and its gradient. Us- 

