A Variational inference and parameter estimation A.1 The empirical Bayes ELBO 19 The expected log-sum-exp in (22) has no closed form. For variational inference, we there- the third term becomes The first and second terms of (20) are straightforward to derive; the third term requires more attention. Using the multinomial logit mass function The first term in (20) is the Shannon entropy of the variational distribution. The second and third terms are (minus) an unnormalized cross entropy – the missing normalization constant is the marginal likelihood. Recall that the variational distribution q .GLYPH<12>$_{1:}$$_{H}$ j GLYPH<22>$_{1:}$$_{H}$ ; 6$_{1:}$$_{H}$ / is a product of normal distributions N .GLYPH<22>$_{h}$ ; 6$_{h}$ / . For empirical Bayes in the MML model, the ELBO objective function (11) becomes In this appendix we describe the variational inference and estimation procedures for the mixed multinomial logit model. A few words on notation: A GLYPH<31> 0 means the matrix A is positive definite; A GLYPH<23> 0 means positive semidefinite; j A j is the determinant of A . For a scalar function s : R ! R and a vector v 2 R $^{n}$, s .v/ means the n GLYPH<2> 1 vector . s .v$_{1}$/; : : : ; s .v$_{n}$ // $^{>}$. H . q / C H ∑ h D 1$^{❊}$ q log p .GLYPH<12>$_{h}$ j GLYPH<16> ; GLYPH<127>/ C H ∑ h D 1 T$_{h}$ ∑ t D 1 ❊ q log p . y$_{ht}$ j x$_{ht}$ ; GLYPH<12>$_{h}$ / : (20) p . y$_{ht}$ j x$_{ht}$ ; GLYPH<12>$_{h}$ / D J ∏ j D 1 [ exp . x > $_{ht j}$GLYPH<12>$_{h}$ / ∑ j 0 exp . x > ht j $_{0}$GLYPH<12>$_{h}$ /$^{]}$$^{y}$$^{j}$ ht ; (21) H ∑ h D 1 T$_{h}$ ∑ t D 1   J ∑ j D 1 y j $_{ht}$. x > $_{ht j}$GLYPH<22>$_{h}$ / GLYPH<0> ❊ q$_{GLYPH<21>}$ log$^{(}$ J ∑ j D 1 exp . x > $_{ht j}$GLYPH<12>$_{h}$ /$^{)}$$^{}$  : (22)