Optimizing HMM log-likelihood with time-dependent prior

Question

I have a HMM (Hidden Markov Model) 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 $\boldsymbol\theta$

Answer 1

The parameter $\theta$ doesn't encode the current state of the HMM and so it isn't time dependent and talking about $\theta$ versus $\theta^{t-1}$ doesn't make sense.

The Baum-Welch algorithm looks for HMM parameters which are maximally likely to have given rise to your observations Z.

We predict maximally likely distributions for $X_t$ and transitions between $X_t$ given the observed data and HMM Parameters and then use these distributions to in turn update our HMM Parameters.

In the Baum-Welch optimisation for HMMs, we distinguish between Observations $Z$, Time-Dependent States $X_t$ and HMM Parameters $\theta$.

If you do in fact want your HMM parameters to be time varying this is a much more complicated problem.