Figure 4: HMC trace plot of log posterior density for four chains. One chain was started at the mode, and the others started at random values. Every 500th iteration is plotted. ing “reverse mode” automatic differentiation, which we discuss more in Section 4.1, the time to evaluate the gradient is about five times the time it takes to evaluate the log posterior, regardless of the number of variables (Griewank and Walther 2008). Therefore, each HMC iteration requires resources that equate to 96 evaluations of the posterior. In other words, the computational cost of 700,000 HMC iterations is equivalent to more than 67 million evaluations of the log posterior. And this assumes that 700,000 iterations were sufficient to collect enough samples from the true posterior. So how much more efficient is our sampling method? We estimated the marginal density q$_{v}$ ( v ) by taking M = 100, 000 proposal draws from an MVN distribution with the mean at the posterior mode q $^{∗}$, and the covariance matrix equal the inverse of the Hessian at the mode, multiplied by a scaling factor of s = 1.3. This value of s is the smallest value for which 0 ≤ F ($_{q}$ | y ) ≤ 1 for all 100,000 samples from g ($_{q}$) . We then collected 300 independent samples in parallel from the target posterior $_{p}$($_{q}$ | y ) . The median number of proposals required for each posterior sample was just under 38,000, the total number of likelihood evaluations was about 16.5 million, and average acceptance rate was 1.8 × 10 − $^{5}$. In absolute terms, the low acceptance rate appears to be unfavorable. However, the total run time is much lower than for MCMC. In our implementation (using a single core on a 2014-vintage Apple Mac Pro), the total time to compute the log posterior density of 1,000 proposal samples is 17 −86200 −86000 −85800 0 200 400 600 Iteration (x 1000) log poster ior density