Figure 2: Quantiles of posterior densities from hierarchical model with conditional conjugacy, as estimated using our method, and with a Gibbs sampler. ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● µ τ 2 σ 2 log posterior − 1.05 − 1.00 − 0.95 − 0.90 0.85 0.90 0.95 1.00 1.05 3.96 3.98 4.00 4.02 4.04 4.06 − 319250 − 319200 − 319150 − 319100 Gibbs Ours Gibbs Ours Gibbs Ours Gibbs Ours and standard deviation s = 2. The q i are normally-distributed across the population, with mean $_{m}$= − 1 and standard deviation $_{t}$= 3. We place uniform priors on m , log s and t . There are 1,503 parameters in this model. This model allows for a conditionally conjugate Gibbs sampler; the steps are described in Gelman et al. (2003, Section 11.7). In this case, the Gibbs sampler is sufficiently fast and efficient, so we have confidence that it does indeed sample from the correct posterior distributions. The Gibbs sampler was run for 2,000 iterations, including a 1,000-iteration burn-in. The process took about five minutes to complete. We then collected 360 independent samples using our method, after estimating q$_{v}$ ( v ) with M = 70, 000 proposals, and applying a scaling factor on the inverse Hessian of 1.02. To get those 360 proposals, we needed 381,507 proposals. Using a single core of a 2014-vintage Mac Pro, this process would take about 23 minutes. However, each draw can be collected in parallel. It took about five minutes to collect all 360 samples when using 12 processing cores. The mean number of proposals for each posterior sample was 1,060 (an acceptance rate of .0009), but the median was only 29. Ten of the 360 samples required more than 10,000 proposals. Figure 2 shows the quantiles of samples in the estimated marginal posterior densities of the population-level parameters m , t $^{2}$, and s $^{2}$, as well as the log of the unnormalized posterior density. We can see that both methods generate effectively the same estimated posterior distributions. In the second example, we model weekly sales of sliced cheese in 88 retail stores. The data are available in the bayesm package for R (Rossi 2012), and were used in Boatwright et al. (1999). Under this model, mean sales for store i in week t has a gamma distribution with mean l it and 13 3.2 Hierarchical model without conditional conjugacy