Bayes objective function is The first line in (25) uses the well-known entropy of the normal distribution. The second line uses the cross-entropy of two normal distributions, also well known. The third line is approximation D0. Here we describe a block coordinate ascent algorithm to maximize (25) over the varia- tional parameters GLYPH<22>$_{1:}$$_{H}$ and 6$_{1:}$$_{H}$ . Although the problem is not jointly convex in all these parameters, each GLYPH<22>$_{h}$ and 6$_{h}$ coordinate update solves a smooth, unconstrained convex opti- mization problem. The requirement 6$_{h}$ GLYPH<23> 0 is satisfied after each update. We initialize the variational parameters at the maximum likelihood estimates from a homogeneous model (in which all agents share a common GLYPH<12> value). The concavity of (25) in GLYPH<22>$_{h}$ follows from the fact that GLYPH<127> GLYPH<31> 0 and from the convexity of the log-sum-exp function. We update GLYPH<22>$_{h}$ using standard algorithms for unconstrained convex optimization (Boyd and Vandenberghe 2004), supplying an analytic gradient and Hessian as follows. Define the function w.GLYPH<22>; 6 ; x / taking values in$_{R}$ $^{J}$, with j th compo- nent exp $^{(}$x > j GLYPH<22> C . 1 = 2 / x > j 6 x$_{j}$ $^{)}$, and normalized to sum to one across j . The gradient of Q L with respect to GLYPH<22>$_{h}$ can then be written 21 A.2 Empirical Bayes variational E-step @ Q L @ GLYPH<22>$_{h}$ D GLYPH<0> GLYPH<127> GLYPH<0> $^{1}$.GLYPH<22>$_{h}$ GLYPH<0> GLYPH<16> / C T$_{h}$ ∑ t D 1 J ∑ j D 1 $^{[}$y j ht GLYPH<0> w $^{j}$.GLYPH<22>$_{h}$ ; 6$_{h}$ ; x$_{ht}$ / $^{]}$x$_{ht j}$ : (26) Q L .GLYPH<22>$_{1:}$$_{H}$ ;6$_{1:}$$_{H}$ I GLYPH<16> ; GLYPH<127>/ D 1 2 H ∑ h D 1 log $^{[}$. 2 GLYPH<25> e / $^{K}$j 6$_{h}$ j ] GLYPH<0> H 2 log $^{(}$. 2 GLYPH<25> / $^{K}$j GLYPH<127> j ) GLYPH<0> 1 2 tr$^{[}$ GLYPH<127> GLYPH<0> 1 H ∑ h D 1 $^{{}$6$_{h}$ C .GLYPH<22>$_{h}$ GLYPH<0> GLYPH<16> /.GLYPH<22>$_{h}$ GLYPH<0> GLYPH<16> / > } ] C H ∑ h D 1 T$_{h}$ ∑ t D 1   J ∑ j D 1 y j $_{ht}$. x > $_{ht j}$GLYPH<22>$_{h}$ / GLYPH<0> log$^{(}$ J ∑ j D 1 exp . x > $_{ht j}$GLYPH<22>$_{h}$ C . 1 = 2 / x > $_{ht j}$6$_{h}$ x$_{ht j}$ /$^{)}$$^{}$  : (25)