The CVB algorithm for the PCFG model keeps the dependencies between the parameters and the hidden parse trees in an exact fashion: We factorise Q(t) by breaking only the weak dependencies between parse trees, while keeping the inside dependencies intact, as we don't make further assumptions about Q(t) for each t. By the above factorisations, we compute the negative variational free energy −F(Q(t)Q(θ|t)) as follows: Maximizing −F(Q(t)Q(θ|t)) requires to update Q(θ|t) and Q(t) in turn.
q(θ|t) is set equal to the true posterior P (θ|w, t, α): Finally, we update the approximate posterior for each parse tree t by using the mean field method in the collapsed space: The inner term P (w i , t i |w ¬i , t ¬i , α) in the above equation is just the unnormalized collapsed Gibbs sampling in (5).