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TV error values for the three inference procedures are presented in Table 1 for the 3- item simulation scenarios, and in Table 2 for the 12-item scenarios. The main conclusion we draw from Tables 1 and 2 is simple: on these data sets, there are no practical differences in accuracy among VEB, VB, and MCMC. The scale of the TV error for all the procedures is the same; that scale is larger in the 12-item case than the 3-item case, but all three procedures exhibit high accuracy on all data sets. The magnitude of our simulation study makes it difficult to carry out the replications required to put standard errors in these tables. But even if the differences among TV errors in every scenario were “significant” under a suitable definition, the patternless alternation in the identity of the most accurate method would make more specific conclusions dubious. 

Figure 1 shows in a different way the comparable accuracy of the three procedures. When there are three choice items, any predictive choice distribution can be plotted as a point in the triangular simplex. The figure shows the close proximity of each procedure’s estimated choice distribution to the true distribution in one simulation scenario. Plots for all the other three-item scenarios are qualitatively the same. Figure 1 also shows contours of the VB and MCMC approximate posterior distributions. We see that VB is producing 

