Evaluating and optimising the cost function

The switching model is a combination of the two models as shown in Section 5.3. Therefore most of the terms of the cost function are exactly as they were in the individual models. The only difference is in the term $C_p(s_k(t)) = \operatorname{E}[ -\log p(s_k(t)\vert \cdots) ]$. The term $\alpha_k(t)$ in Equation (6.31) changes to $\alpha_{k,i}(t)$ for the case $M_t = i$ and has the value

$$\begin{split}\alpha_{k,i}(t) &= \operatorname{E}\left[ (s_k(t... ...s}(t-1))} {\partial s_k(t-1)} \widetilde{s}_k(t-1). \end{split}$$ (6.41)

With this result the expectation of Equation (6.33) involving the source prior becomes

$$\begin{split}C_p(\boldsymbol{S}) &= \operatorname{E}\left[ -\... ...tilde{v}_{M_i,k} - 2 \overline{v}_{M_i,k}) \right). \end{split}$$ (6.42)

The update rules will mostly stay the same except that the values of the parameters of $v_{m_k}$ must be replaced with properly weighted averages of the corresponding parameters of $v_{M_i,k}$. The HMM prototype vectors will be taught using $\mathbf{s}(t)- \mathbf{g}(\mathbf{s}(t-1))$ as the data.