content_image/324146.jpg content_image/324145.jpg
Bayes objective function is 

<img src='content_image/324146.jpg'>

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. 

## A.2 Empirical Bayes variational E-step 

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 

<img src='content_image/324145.jpg'>

