its variability does not affect even the least significant digit of our reported results. We handle the integral over GLYPH<12> in the same way for the estimated predictive choice distributions furnished by each of the three inference procedures. For VEB, the estimate is (16), with GLYPH<16> and GLYPH<127> replaced by O GLYPH<16>$_{VEB}$ and O GLYPH<127>$_{VEB}$ . On the other hand, with VB and MCMC, we obtain a posterior distribution over GLYPH<16> and GLYPH<127> ; we take the mean of (16) under this posterior as a point estimate of the predictive choice distribution: For VB, the posterior density p .GLYPH<16> ; GLYPH<127> j D / in (18) is approximated by the fitted variational distribution For MCMC, the posterior is approximated as usual by the empirical distribution of draws from a Markov chain. In both cases, we handle the integral over GLYPH<16> and GLYPH<127> in (18) with another exhaustive Monte Carlo approximation. We measure the error of each inference procedure as the distance from its estimate of the predictive choice distribution to the true distribution. As the metric on distributions, we use total variation (TV) distance, leading to what we call the “TV error” of each procedure. In this setting, TV error equals the maximum, over all choice-item subsets, of the difference in the probabilities assigned to the subset by the estimated versus the true choice distribution. We also need to choose the attribute matrix x$_{new}$ at which the true and estimated predictive choice distributions are calculated. In each simulation scenario, we computed the TV error of VEB on 25 different random draws of x$_{new}$ . We then compared the three procedures using the x$_{new}$ which yielded the median of the 25 TV errors. In this sense our results are representative of accuracy for a “typical” item attribute matrix. However, results using any one of the 25 matrices were qualitatively the same in the examples we checked. 13 O p . y$_{new}$ j x$_{new}$ ; D / D ∫ [∫ p . y$_{new}$ j x$_{new}$ ; GLYPH<12> / p .GLYPH<12> j GLYPH<16> ; GLYPH<127>/ d GLYPH<12>$^{]}$ p .GLYPH<16> ; GLYPH<127> j D / d GLYPH<16> d GLYPH<127> : (18) q $^{(}$GLYPH<16> j GLYPH<22>$_{GLYPH<16>}$ ; 6$_{GLYPH<16>}$ $^{)}$q $^{(}$GLYPH<127> j 7 GLYPH<0> $^{1}$; ! ) : (19)