Finding optimal $q(\boldsymbol {M})$

Assuming $q(\boldsymbol {\theta })$ is fixed, the cost function can be written, up to an additive constant, in the form

$$\begin{split}C(\boldsymbol{M}) &= \sum_{\boldsymbol{M}} q(\bo... ...A} + \sum_{t=1}^T C(\mathbf{x}(t) \vert M_t) \bigg] \end{split}$$ (6.11)

where $C(\mathbf{x}(t) \vert M_t)$ is the value of $\operatorname{E}[ p(\mathbf{x}(t) \vert M_t, \boldsymbol{\theta}) ]$, i.e. the ``cost'' of current data sample given the HMM state.

By defining

$$\pi_i^* = \exp\left(\int q(\boldsymbol{\pi}) \log \pi_{i} d\boldsymbol{\pi}\right) a_{ij}^* = \exp\left(\int q(\mathbf{A}) \log a_{ij} d\mathbf{A}\right),$$ (6.12)

Equation (6.11) can be written in the form

$$C_{q(\boldsymbol{M})} = \sum_{\boldsymbol{M}} q(\boldsymbol{M}) \... ...1}}^* \right] \left[ \prod_{t=1}^T \exp(- C(\mathbf{x}(t) \vert M_t)) \right]}.$$ (6.13)

The expression $\int q(x) \log \frac{q(x)}{p^*(x)}$ is minimised with respect to $q(x)$ by setting $q(x) = \frac{1}{Z} p^*(x)$ where $Z$ is the appropriate normalising constant [39]. This can be proved with similar reasoning as in Equation (3.13).

The cost in Equation (6.13) can thus be minimised by setting

$$q(\boldsymbol{M}) = \frac{1}{Z_M} \pi_{M_1}^* \left[ \prod_{t=1}^... ...t+1}}^* \right] \left[ \prod_{t=1}^T \exp(- C(\mathbf{x}(t) \vert M_t)) \right]$$ (6.14)

where $Z_M$ is the appropriate normalising constant.

The derived optimal approximation is very similar in form to the exact posterior in Equation (4.6). Therefore the point probabilities of $q(M_1 = i)$ and $q(M_t = j \vert M_{t-1}=i)$ can be evaluated with a modified forward-backward iteration. The result is the same as in Equation (4.9) except that in the iteration, $\pi_i$ is replaced with $\pi_i^*$, $a_{ij}$ is replaced with $a_{ij}^*$ and $b_i(\mathbf{x})$ is replaced with $\exp(-C(\mathbf{x}(t) \vert M_t=i))$.