reports the details behind the extension; here we summarize the main ideas. Although the joint posterior (7) is not factorized, we continue to use a family Q of factorized distributions for the variational approximation: Using a factored family for a non-factored posterior is commonly called mean-field varia- tional inference. In (14), q $^{(}$GLYPH<16> j GLYPH<22>$_{GLYPH<16>}$ ; 6$_{GLYPH<16>}$ ) is a K -variate normal density; q $^{(}$GLYPH<127> j 7 GLYPH<0> $^{1}$; ! ) is an inverse Wishart density; and the q .GLYPH<12>$_{h}$ / factors are K -variate normal densities as before. In the analysis and the algorithm, it is convenient to use a well-known equivalence, treating q $^{(}$GLYPH<127> j 7 GLYPH<0> $^{1}$; ! ) as a Wishart distribution W .7; !/ on GLYPH<127> GLYPH<0> $^{1}$. We are therefore optimizing over variational parameters GLYPH<21> : D $^{{}$GLYPH<22>$_{GLYPH<16>}$ ; 6$_{GLYPH<16>}$ ; 7; ! ; GLYPH<22>$_{1:}$$_{H}$ ; 6$_{1:}$$_{H}$ $^{}}$. The variational problem for hierarchical Bayes is to find the best approximating distribution q .GLYPH<12>$_{1:}$$_{H}$ ; GLYPH<16> ; GLYPH<127> j GLYPH<21> $^{GLYPH<3>}$/ in Q by solving the analog of (13): As with empirical Bayes, we use a block coordinate ascent optimization algorithm to solve (15), iterating through the coordinate blocks that define GLYPH<21> . Here again, all coordi- nate updates are convex optimizations. The details appear in Appendix A: updates for GLYPH<22>$_{GLYPH<16>}$ , 6$_{GLYPH<16>}$ , 7 all have simple closed forms; ! has a closed form which requires no updating; and the GLYPH<22>$_{h}$ and 6$_{h}$ updates are similar to the empirical Bayes case. We compared the accuracy and speed of the variational methods described in the previous section to a standard and widely used MCMC approach (Allenby and Rossi 2003), on a suite of simulated data sets. Each data set was generated using the discrete choice model given by (3) and (4). To simulate a data set with J choice items, K item attributes, and 10 4 Empirical results Q 3 q .GLYPH<12>$_{1:}$$_{H}$ ; GLYPH<16> ; GLYPH<127> j GLYPH<21>/ : D q $^{(}$GLYPH<16> j GLYPH<22>$_{GLYPH<16>}$ ; 6$_{GLYPH<16>}$ $^{)}$q $^{(}$GLYPH<127> j 7 GLYPH<0> $^{1}$; ! ) H ∏ h D 1 q .GLYPH<12>$_{h}$ j GLYPH<22>$_{h}$ ; 6$_{h}$ / : (14) GLYPH<21> GLYPH<3> D argmax GLYPH<21> 2 3 L .GLYPH<21>/ : (15)