# Optimizing this log-likelihood

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 $$\boldsymbol\theta$$