The top-level parameters GLYPH<16> and GLYPH<127> , to be estimated by maximum marginal likelihood, rep- resent the distribution of attribute preferences across the population. In particular, GLYPH<127> gives us information about the correlation of preferences between agents. A fully Bayesian approach requires hyperprior distributions for GLYPH<16> and GLYPH<127> . As is standard, we use conditionally conjugate distributions: In (5), GLYPH<12>$_{0}$ and GLYPH<127>$_{0}$ are pre-specified hyperparameters; W GLYPH<0> $^{1}$. S GLYPH<0> $^{1}$; GLYPH<23> / is the inverse Wishart distribution with scale matrix S GLYPH<0> 1 and GLYPH<23> degrees of freedom; and S and GLYPH<23> are hyperpa- rameters fixed in advance. We call the fully Bayesian approach to MML model inference “hierarchical Bayes.” We have presented the component hierarchical distributions in the mixed multinomial logit model. Now we turn to the question of estimation and inference procedures. In the follow- ing, variable names inside of p . GLYPH<1> / are used to distinguish among densities: we denote the pdfs in (3)–(5) by p . y$_{ht}$ j x$_{ht}$ ; GLYPH<12>$_{h}$ / , p .GLYPH<12>$_{h}$ j GLYPH<16> ; GLYPH<127>/ , p .GLYPH<16> j GLYPH<12>$_{0}$ ; GLYPH<127>$_{0}$ / , and p .GLYPH<127> j S ; GLYPH<23> / , respec- tively. We let D D f x$_{ht}$ ; y$_{ht}$ g denote all observed variables, i.e., the data. For the empirical Bayes version of the MML model, the posterior density of the latent preference vectors, p .GLYPH<12>$_{1:}$$_{H}$ j D ; GLYPH<16> ; GLYPH<127>/ , is The joint posterior density for hierarchical Bayes, p .GLYPH<12>$_{1:}$$_{H}$ ; GLYPH<16> ; GLYPH<127> j D / , is 6 p .GLYPH<16> / p .GLYPH<127>/ ∏ H h D 1 p .GLYPH<12>$_{h}$ j GLYPH<16> ; GLYPH<127>/ ∏ T$_{h}$ t D 1 p . y$_{ht}$ j x$_{ht}$ ; GLYPH<12>$_{h}$ / ∫ p .GLYPH<16> / p .GLYPH<127>/ ∏ H h D 1 ∫ p .GLYPH<12>$_{h}$ j GLYPH<16> ; GLYPH<127>/ ∏ T$_{h}$ t D 1 p . y$_{ht}$ j x$_{ht}$ ; GLYPH<12>$_{h}$ / d GLYPH<12>$_{h}$ d GLYPH<16> d GLYPH<127> : (7) H ∏ h D 1 p .GLYPH<12>$_{h}$ j GLYPH<16> ; GLYPH<127>/ ∏ T$_{h}$ t D 1 p . y$_{ht}$ j x$_{ht}$ ; GLYPH<12>$_{h}$ / ∫ p .GLYPH<12>$_{h}$ j GLYPH<16> ; GLYPH<127>/ ∏ T$_{h}$ t D 1 p . y$_{ht}$ j x$_{ht}$ ; GLYPH<12>$_{h}$ / d GLYPH<12>$_{h}$ : (6) GLYPH<16> j GLYPH<12>$_{0}$ ; GLYPH<127>$_{0}$ GLYPH<24> N$_{K}$ .GLYPH<12>$_{0}$ ; GLYPH<127>$_{0}$ / ; GLYPH<127> j S ; GLYPH<23> GLYPH<24> W GLYPH<0> $^{1}$. S GLYPH<0> $^{1}$; GLYPH<23> /: (5) 3 Variational inference for the MML model