The Dirichlet has the density function: Thus the prior overθ is a product of Dirichlets,which is conjugate to the PCFG likelihood function .
the posterior P (θ|s, t, α) is also a product of Dirichlets, also factoring into a Dirichlet for each nonterminalN , where the parameters αr are augmented by the number of times rulē r is observed in tree t: We can see that αr acts as a pseudocount of the number of timesr is observed prior to t. To make use of this prior, we use the Variational Bayes (VB) technique for PCFGs with Dirichlet Priors presented by Kurihara and Sato (2004).