H agents, we first fixed values of GLYPH<16> and GLYPH<127> , the parameters controlling the distribution of preferences in the agent population. We then independently drew a GLYPH<12>$_{h}$ vector for each agent, according to (4). We also drew for each agent 25 iid J GLYPH<2> K item attribute matrices x$_{ht}$ consisting of iid N . 0 ; 0 : 5$^{2}$/ entries. Finally, for each agent, we used x$_{ht}$ and GLYPH<12>$_{h}$ to simulate 25 choice events y$_{ht}$ , according to (3). Thus, in our data sets, each agent has 25 observed choices. We simulated a total of 32 different scenarios by varying J , K , H , and the selection of GLYPH<16> and GLYPH<127> . Specifically, each data set corresponds to a distinct configuration of the following candidate values: 3 or 12 choice items J ; 3 or 10 item attributes K ; 250, 1000, 5000, or 25000 agents H ; and “low” or “high” heterogeneity of the agent population. In the low- heterogeneity scenario, the K GLYPH<2> 1 vector GLYPH<16> consists of evenly spaced values from -2 to 2, and the K GLYPH<2> K matrix GLYPH<127> is 0.25 times the identity matrix. In the high-heterogeneity scenario, GLYPH<16> is the same, but GLYPH<127> is the identity matrix. The data sets with high heterogeneity have much more diverse collections of preference vectors GLYPH<12>$_{h}$ . We ran variational empirical Bayes (VEB), variational hierarchical Bayes (VB), and the standard MCMC algorithm on the observable data from each of the 32 simulation sce- narios. For VEB, we declared convergence as soon as an E-step/M-step iteration caused the parameter estimates’ joint Euclidean norm to change by less than 10 GLYPH<0> $^{4}$, relative to their norm at the beginning of the iteration (here we mean the joint norm of all the variational parameters, together with the model parameter estimates O GLYPH<16> and O GLYPH<127> ). The convergence crite- rion for VB was the same, except GLYPH<16> and GLYPH<127> do not have point estimates – we look instead at the change in the variational parameters corresponding to their posterior approximation, namely GLYPH<22>$_{GLYPH<16>}$ , 6$_{GLYPH<16>}$ , and 7 . Choosing MCMC convergence criteria is more delicate. We tried to set the number of burn-in iterations and the thinning ratio algorithmically, using the technique of Raftery and Lewis (1992). But on several of our data sets, typical control parameter values, such as 11