Maximum likelihood estimation (MLE) for Markov Chain initial distribution?

Question

I am working on using MLE to estimate a Markov Chain, I have successfully estimated the transition matrix $A$, using the method provided in http://www.stat.cmu.edu/~cshalizi/462/lectures/06/markov-mle.pdf

Which is simply counting $A_{i,j} = N_{i,j}/\sum N_{i,j}$

But is there a way to estimate the initial distribution $\pi$ or named $\mu^{0}_{i}$s ?

It seems like MLE can do it but I dont know how.

Answer 1

Sure, one needs to run the Markov chain several times, then the MLE of $\pi(i)=\mu^0_i$ is the proportion of times the process started at state $i$.