Predicting a Markov chain next state using previously predicted states

Suppose we have a Markov chain with two states A and B. This associated transition matrix is:

$$$$P_{mc}= \begin{pmatrix} 0.3 & 0.7\\ 0.6 & 0.4 \end{pmatrix}$$$$

This matrix is empirical and computed from a set of observations:

e.g :A->B->A->A->A->B->A->A->B->B->A->B->B->A->B->A->B->A->B->B->B

My question is, After predicting a new state (B or A) how accurate is it to generate a new transition matrix P_mc based on the new sequence and is there any theoretical limits to doing this?

• I don't quite understand what the new matrix' generation is based on - are you updating the matrix based on a new observation, or the actual prediction? – jkm Feb 21 at 10:57
• The matrix is updated using the actual prediction – nidabdella Feb 21 at 10:59

For example, an initial transition matrix $$[[0.99, 0.01], [0.01, 0.99]]$$ will asymptotically wind up predicting either only As or only Bs, depending on your start state.