shape r$_{i}$ . The mean is a log-linear function of price, and the percentage of “all category volume” on display in that week. The prior on each r$_{i}$ is a half-Cauchy distribution with a scale parameter of 5, and the prior on each b i is MVN with mean m and covariance W . The hyperparameters m and W have weakly informative MVN and inverse Wishart priors, respectively. There are 361 parameters in this model. This model does not allow for a conditionally conjugate Gibbs sampler. Instead, we use Hamil- tonian Monte Carlo (HMC, Duane et al. 1987; Neal 2011), which uses the gradient of the log pos- terior to simulate a dynamic physical system. We selected HMC mainly because it is known to generate successive samples that are less serially correlated than draws that one might sample using other MCMC methods. We implemented HMC using the Stan software package (Stan Development Team 2014). We ran 5 parallel chains for 800 iterations each, and discarding the first half of the draws. The chains appeared to display little autocorrelation, so we have confidence that the HMC samples form a good estimate of the true posterior distributions. We then estimated the model using our method with different numbers of proposal draws to estimate q$_{v}$ ( v ) , and different scale factors on the covariance of the proposal density. In Figure 3, we compare the estimates densities for elements of m , the baseline and marginal effects on sales. We see that even with a relatively coarse estimate of q$_{v}$ ( v ) , and an overly diffuse proposal density, our method generates estimates of the posterior densities that are not only comparable with each other, but also comparable those generated by Stan. The acceptance rates across different runs using our method was .0003. It took about 1.4 seconds to sample and evaluate a block of 1,000 proposals on a single processing core. Our third example concerns a non-conjugate heterogeneous hierarchical model, in which some parameters are only weakly identified. The data structure is described in Manchanda et al. (2004); we use the simulated data that are available in the bayesm package for R (Rossi 2012). In this dataset, for each of 1,000 physicians, we observe weekly prescription counts for a single drug 3.3 Model with weakly identified parameters 14 log l it = b i $_{1}$+$_{b}$$_{i}$$_{2}$ log PRICE$_{it}$ +$_{b}$$_{i}$$_{3}$ DISP$_{it}$ (16)