Sorry, my bad. Yes, you are right. It should be $\boldsymbol{\pi}$ I fixed the post. Thank you.

Thank you. I understand the concern and I myself also doubt my approach. Since I have many states and some transitions are just like 9 or 10 out of millions of total transitions, it comes to the problem of having many parameters to estimate, so restricting some to be 0 can reduce the number a bit. You are right that I am heading to some regularization of the estimates: where to place zeros on the probability transition matrix makes the most sense. There will be some schemes of penalization, but I think in the first step, I should make sure I can obtain the log-likelihood value sensibly.

Thank you. But I think I need to add a little bit more. There are some transitions that did occur but very rarely so the probability estimates came out very low so I want to try constraining them to be exactly 0. I want to obtain the log-likelihood value because I want to compare one setup where some entries are 0 to another setup where some other entries are 0. Please let me know if I can do anything about it. Thank you.