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## Main Question or Discussion Point

**[SOLVED] Markov chain problem**

Hi, all! This is not a homework thing or something like that. I'm just curious.

Suppose a Markov chain p(n+1)=Q*p(n), where Q=[q11, q12; q21, q22] is the transition probability matrix and p(n)=[p1(n);p2(n)] is the probability vector.

Given Q and p(1), we can generate a realization from it, i.e., a sequence of ones and zeros.

Now I want to do the inverse. Given a long enough sequence of ones and zeros, Q can be obtained by counting the number of changes from zero to zero(which will be q11), zero to one(q12), one to zero(q21), and one to one(q22). But, how can I get the initial probability vector p(1) from this sequence? Maximum Likelihood Estimation, blah blah? I don't know about that. Please, someone give me a clue how I can do this. Or is it impossible to do that from just only one realization sequence?

Thanks in advance.