Hidden Markov Model and volatile Matrix A Suppose that we have a Hidden Markov Model (HMM) in which my Transition Distribution (also known as Matrix A) between hidden states dramatically change during the time; However, my Emission Distribution (also known as Matrix B) still the same during all times.
How can I possibly handle the volatile behavior of Matrix A during time?
By the way, I have fully observed the hidden states. Hence, I compute Matrix A, by counting the transitions between different states.
I would be very thankful if someone could possibly introduce me a similar paper work or give me a hint on how to handle this intermittence of matrix A during time?
Thank you very much :)
 
 A: HMM is a special form of probabilistic graphical model that has many constraints. Both transition probability matrix and emission probability matrix do not change over time are two important constraints to make the into HMM. If you want to make transition matrix change over time, then it is not a standard HMM.
But first, I would like to ask why to do so. We want to do such revisions is coming from domain knowledge or data? I can only think it comes from domain knowledge. Because the uncertainties in the hidden states should be captured by the probabilistic transition matrix and making the transition matrix change over time, you have "too much freedom" in the model.
If the notion of transition matrix changes over time come from domain knowledge, then may be constructing different HMMs on different time period would be a solution.
In sum, when we talk about a model, we may need to think how much free parameters in there and do we have sufficient data to reliably estimate these parameters. A matrix changes over time would cause too many free parameters and may not be useful in real world. (suppose we have $\Omega$ hidden stages, the transition matrix has $\Omega \times (\Omega-1)$ free parameters, prob need to sum to $1.0$, in addition, if sequence length is $N$ then there are $\Omega \times (\Omega-1) \times N$ free parameters), and we are not counting the free parameters in emission matrix. Such level of complexity is not practical / parameters cannot reliably estimated in real world with limited data.
