How would I convert this 4x4 transiton matrix to a 2x2 transition matrix while maintaing that all rows sum to 1

Question

Say I have a transition matrix with 4 states labeled 1 to 4 as follows:

matrix = [[.25,.25,.5,0],
          [ 0,.25,.5,.25],
          [.25,.25,.25,.25],
          [.25,.25,0, .5 ]]

Say I want to convert this transition matrix to one with only 2 states, good and bad, where states 1 and 2 from the original matrix are part of the "good" state and states 3 and 4 are part of the "bad" state.

How would I convert the original transition matrix to a 2x2 matrix, while keeping the properties that the rows sum to 1 (transition matrix property).

Adding all the transitions doesn't work. For example the ways of going from "good" to "good" would be summing up going from state 1 to 1, 1 to 2, 2 to 1, and 2 to 2 but that's .75 and if you do the same calculate for "good" to "bad" the row ends up summing to 2 instead of 1.

Thank you.

Answer 1

Denote by X the Markov chain with state space {1,2,3,4} given by this transition matrix, and denote by Y the new Markov chain with state space {good, bad}.

P(Y2 = good | Y1 = good) = P(X2 = 1 or 2 | X1 = 1 or 2) = P(X2 = 1 or 2 & X1 = 1 or 2) / P(X1 = 1 or 2)

The transition matrix only provides conditional probabilities P(X2 = j | X1 = i). So you can't get the transition matrix of Y if you only have the one of X. You need the distribution of X, which can be obtained from the distribution of the initial value.