Finding optimal $q(\boldsymbol {\theta })$ for Dirichlet parameters

Let us now assume that $q(\boldsymbol {M})$ is fixed and optimise $q(\mathbf{A})$ and $q(\boldsymbol{\pi})$. Assuming that everything else is fixed, the cost function can be written as a functional of $q(\mathbf{A})$, up to an additive constant

$$\begin{split}C(\mathbf{A}) &= \int q(\mathbf{A}) \bigg[ \log ... ...{\prod_{i,j=1}^N a_{ij}^{(W_{ij} - 1)}} d\mathbf{A} \end{split}$$ (6.15)

where $W_{ij} = u^{(A)}_{ij} + \sum_{t=1}^{T-1} q(M_{t}=i, M_{t+1}=j)$.

As before, the optimal $q(\mathbf{A})$ is of the form $q(\mathbf{A}) = \frac{1}{Z_A} a_{ij}^{(W_{ij} - 1)}$. The update rule for the parameters $\hat{a}_{ij}$ of $q(\mathbf{A})$ is thus

$$\hat{a}_{ij} \leftarrow W_{ij} = u^{(A)}_{ij} + \sum_{t=1}^{T-1} q(M_{t+1}=j \vert M_{t}=i).$$ (6.16)

Similar reasoning for $\boldsymbol{\pi}$ gives the update rule

$$\hat{\pi}_i \leftarrow u^{(\pi)}_i + q(M_1 = i).$$ (6.17)