first-order conditions for block coordinate ascent lead to the updates 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 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. Let f .v/ be a function from ❘ K to ❘ . According to the multivariate delta method for moments (Bickel and Doksum 2007), Consider the case B An application of the delta method ! GLYPH<23> C H ; (41) 7 ( S GLYPH<0> 1 C H ∑ h D 1 $^{(}$6$_{h}$ C .GLYPH<22>$_{GLYPH<16>}$ GLYPH<0> GLYPH<22>$_{h}$ /.GLYPH<22>$_{GLYPH<16>}$ GLYPH<0> GLYPH<22>$_{h}$ / > $^{)}$C H 6$_{GLYPH<16>}$$^{)}$$^{GLYPH<0>}$$^{1}$ : (42) ❊ f . V / GLYPH<25> f . ❊ V / C 1 2 tr$^{[(}$ @ f . ❊ V / @ v@ v > ) Cov . V /$^{]}$ : (43) f .v/ D log $^{(}$1 $^{>}$exp . x v/ ) ; (44) 25 GLYPH<22>$_{GLYPH<16>}$ $^{(}$GLYPH<127> GLYPH<0> 1 0 C H !7 ) GLYPH<0> 1 ( GLYPH<127> GLYPH<0> 1 0 GLYPH<12>$_{0}$ C !7 H ∑ h D 1 GLYPH<22>$_{h}$$^{)}$ ; (39) 6$_{GLYPH<16>}$ $^{(}$GLYPH<127> GLYPH<0> 1 0 C H !7 ) GLYPH<0> 1 : (40)