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.
lets call your matrix $A=[a_{ij}]$, and assume $Av = v = [v_i]$ for some nonnegative definite $v$ whose entries sum to $1$ ie its the stationary probabilities for the states.
The best way to come at your problem is to use $v$ to work out what the transitions are from a 'good' state ie state 1 or 2. So, given that you're in 1 or 2, you have a probability of $v_1/(v_1 + v_2)$ of being in state 1, and you can work out the probability of being in state 2. The row that you might write down for transitions from 'good' would be $$ \frac{v_1}{v_1 + v_2}[a_{1i}] + \frac{v_2}{v_1 + v_2}[a_{2i}] $$ and then, by summing the first pair of entries, and the last pair, you have a row of the matrix for transitions between good and bad states.
Hope that's relatively clear.