I have a HMM which emits an observation Z.

The parameters of the HMM are $\boldsymbol\theta$. $$\boldsymbol\theta = {\boldsymbol{A},\boldsymbol{B},\pi}$$

Where $\boldsymbol{A}$ is the transition matrix, $\boldsymbol{B}$ is the emission matrix and $\boldsymbol{\pi}$ is the initial distribution matrix over the states.

I want to maximise $$= \sum_{Z} q(Z)log P(Z | \boldsymbol{\theta})$$

where $q(Z)=p(Z|\boldsymbol\theta^{t-1})$ is a probability of the particular Z being observed. Where the $\boldsymbol\theta^{t-1}$ is the HMM parameter of the previous step.

I know that I can maximise $$= \sum_{Z} log P(Z | \boldsymbol{\theta})$$ using Baum-Welch algorithm but is there a good way to maximise the expression above?

Kindly note that q(Z) remains fixed in this problem and we are maximising the function wrt $\boldsymbol\theta$


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