# How to calculate a 𝑛-step transitions of a Discrete-time Markov Chain for Figure 17.1 (b) in book “Machine Learning - A Probabilistic Perspective”

chapter 17 of the book "Machine Learning - A Probabilistic Perspective" gives this figure

which is the probability of getting from i to j in exactly n steps. Obviously A(1) = A.

In the case of Figure 17.1 (b), the probability of getting from node 1 to node 3 in exactly 2 steps is

$$A_{13}(2) = A_{12}(1)A_{23}(1)$$

Is that correct?

• Although it's correct, it's not very illuminating. The nature of the situation starts becoming apparent when you increase the number of steps. – whuber Jul 25 at 10:57
• You can get $n$-step transitions from discrete markov chain by exponentiating the transition matrix – logistic Jul 25 at 13:49
• @logistic do you mean this. $A(n)=A^n$ ? – czlsws Jul 26 at 0:04
• Yes @czlsws that's what I mean – logistic Jul 26 at 12:42
• @logistic thanks for your comments. And how to apply that on "Figure 17.1 (b) node 1 to node 3" – czlsws Jul 26 at 22:14