When v 0, f (v) ≈ v − 0.5, so roughly speaking, VB for multinomials involves adding α−0.5 to the expected counts when they are much larger than zero, where α is the Dirichlet prior parameter.
vB can be viewed as a more principled version of the wellknown ad hoc technique for approximating Bayesian estimation with EM that involves adding α−1 to the expected counts.