Markov chain: join states in Transition Matrix

Question

I need to merge two state in transition Matrix:

For example: i have the matrix below

              A       B       C      D      E      F 
          A  0.5     0.4      0      0      0.1    0

          B  0.5     0.1      0.2    0.1    0.1    0

          C  0       0.1      0.9    0      0      0

          D  0       0        0      0.7    0.3    0

          E  0       0.2      0      0.7    0      0.1

          F  0       0        0      0.5    0      0.5

And i want to join the states D and E:

              A      B        C    (D+E)    F 
          A  0.5     0.4      0      ?      0

          B  0.5     0.1      0.2    ?      0

          C  0       0.1      0.9    ?      0

       (D+E) ?       ?        ?      ?      ?

          F  0       0        0      ?      0.5

what are the formulas to obtain the row and column (D+E)?

Using the constraint: "the sum over column must be equal to 1" is simple to calculate the elements:

(A,(D+E))=0.2

(B,(D+E))=0.2

(C,(D+E))=0.1

(F,(D+E))=0.5

how can I calculate the elements of row ((D+E),i)?

I am not sure you are asking the question in the right place. Stackoverflow is for programming and it seems to me you are asking about pure theory here. Maybe Mathematics SE would be a better place for this question. — Chop

Robert Dodier Robert Dodier · Accepted Answer · 2015-06-08T18:57:18

I think you can work it out by writing P(A | D + E) = P(A, D + E) / P(D + E) and then applying de Morgan's law and noting D and E are mutually exclusive. I get P(A | D + E) = (P(A | D) P(D) + P(A | E) P(E)) / (P(D) + P(E)), likewise for any other states. The marginal probabilities just the elements of the eigenvector with eigenvalue 1. Disclaimer: you'll want to verify this.

Markov chain: join states in Transition Matrix

2 Answers