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. $^{2}$Shorter path lengths were less efficient, and longer ones frequently jumped so far from the regions of high posterior mass that the computation of the log posterior would underflow. We had the same problem with the No U-Turn Sampler (Hoffman et al. 2014), whether using Stan, or coding the algorithm ourselves. The HMC extensions in Girolami and Calderhead (2011) are inappropriate for this problem because the Hessian is not guaranteed to be positive definite for all parameter values. 16 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- Table 1: Effective sample sizes for estimates of marginal posterior distributions of population-level param- eters, using the final 100,000 samples of HMC chains. Chain Chain 1 2 3 4 1 2 3 4 D$_{1,1}$ 295 250 266 267 g 1 3 4 6 2 D$_{1,2}$ 16 24 27 29 g 2 10 33 22 22 D$_{1,3}$ 6 12 26 14 g 3 3 15 3 3 D$_{1,4}$ 51 43 43 60 Chol ( V )$_{1,1}$ 553 513 500 511 D$_{2,1}$ 4703 5933 5981 1444 Chol ( V )$_{2,1}$ 7844 18761 13796 9852 D$_{2,2}$ 507 573 526 538 Chol ( V )$_{2,2}$ 530 523 517 477 D$_{2,3}$ 407 445 400 407 mean( b 1 $_{i}$) 6 5 12 12 D$_{2,4}$ 3708 4797 3680 4926 mean( b 2 $_{i}$) 21 22 28 10