content_image/320650.jpg content_image/320651.jpg content_image/320645.jpg content_image/320646.jpg content_image/320647.jpg content_image/320648.jpg
first-order conditions for block coordinate ascent lead to the updates 

<img src='content_image/320650.jpg'>

<img src='content_image/320651.jpg'>

By inspection, 6$_{GLYPH<16>}$ GLYPH<23> 0, so this constraint need not be explicitly enforced. Note the similar- ity to conjugate posterior updating: on the precision scale, 6$_{GLYPH<16>}$ is the sum of the prior pre- cision matrix GLYPH<127> GLYPH<0> 1 0 and H copies of the variational posterior mean !7 for GLYPH<127> GLYPH<0> $^{1}$. Similarly, GLYPH<22>$_{GLYPH<16>}$ is a precision-weighted convex combination of the prior vector GLYPH<12>$_{0}$ and the empirical average of the variational posterior means GLYPH<22>$_{1:}$$_{H}$ for GLYPH<12>$_{1:}$$_{H}$ . 

The updates for 7 and ! are similarly straightforward to derive; we obtain 

<img src='content_image/320645.jpg'>

<img src='content_image/320646.jpg'>

Notice that the solution (41) for ! involves only the constants GLYPH<23> and H . We compute ! once in advance, leaving it unchanged during the variational optimization. 

## B An application of the delta method 

Let f .v/ be a function from ❘ K to ❘ . According to the multivariate delta method for moments (Bickel and Doksum 2007), 

Consider the case 

<img src='content_image/320647.jpg'>

<img src='content_image/320648.jpg'>

