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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{displaymath}\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}\end{displaymath} (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

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

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

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

Antti Honkela 2001-05-30