where is the inside probability of observation w i,k = w i , ..., w k given A is the root of the subtree, and P OUT (A, i, k) is the probability of A spanning (i, k), together with the rest of w. In the M step, we find the optimal θ based on the MLE principle: VB inference is the generalisation of EM in the sense that it allows arbitrary parametric forms of Q(θ).
the update equation in the M step is: By the conjugacy property, the new Q(θ) is still in Dirichlet distribution form except with updated hyperparameters as shown by Kurihara and Sato (2006).