not only a posterior mean similar to MCMC, but also a similar posterior density in the neighborhood of the mean. For each simulated data set, we ran the three procedures in turn on the same unloaded machine: a 64-bit dual-core 3.2 GHz Intel Xeon processor with 8 GB of main memory. For the MCMC inference, we used the rhierMnlRwMixture function in the R package bayesm (Rossi and McCulloch 2007), which has efficient vectorized implementations of the inner sampling routines. This package stores all MCMC draws in memory, however. For our largest data sets, with 25,000 decision-makers, the machine’s memory was ex- hausted before MCMC converged. We were able to run MCMC for 1,000 iterations in this case, which allowed us to extrapolate accurately the time that would have been required for 6,000 iterations. We implemented the variational algorithms in R, with compiled C components for the numerical optimization routines. Figure 2 displays time to convergence on each data set for the three procedures, accord- ing to the convergence criteria previously described. Within each panel, convergence time is plotted as a function of the number of agents, for fixed values of the other simulation parameters. Note that the vertical axis shows convergence time on a logarithmic scale, to ease comparison of MCMC to the variational methods. All the procedures scale roughly linearly with the number of agents, which leads to the logarithmic curves seen in the figure. The conclusions are the same in all the scenarios we simulated: variational methods con- verge faster than MCMC, and the magnitude of the difference increases with the number of observed agents. MCMC uses two days of computation time for 25,000 agents with 12 choice items, 10 item attributes, and high heterogeneity, versus an hour for each of the variational techniques. In the same setting but with low heterogeneity, MCMC’s two-day computation compares with two hours for VEB and six hours for VB. In other scenarios, variational run times are measured in minutes, as opposed to hours or days for MCMC. 16 4.2 Speed