How do you construct a transition matrix from a Markov chain state diagram?

Conditions

• The Markov chain has a finite number of states.
• The state diagram shows all possible transitions and their probabilities.

Reasoning, step by step

1. Identify the states of the Markov chain from the diagram.
2. Create an $N \times N$ matrix where $N$ is the number of states.
3. For each state $i$ (row) and state $j$ (column), find the transition probability from $i$ to $j$ in the diagram.
4. Fill in the matrix entry $P_{ij}$ with this probability.
5. Verify that the sum of the probabilities in each row equals 1.

Example

For a two-state chain with states A and B, the diagram shows A->$A (0.3)$, A->$B (0.7)$, B->$A (0.4)$, and B->$B (0.6)$. The matrix is constructed as $P = \begin{bmatrix} 0.3 & 0.7 \\ 0.4 & 0.6 \end{bmatrix}$. The speaker checks that $0.3 + 0.7 = 1$ and $0.4 + 0.6 = 1$.

Common misconceptions

• Placing the transition probability from $i$ to $j$ in row $j$ and column $i$ (transposing the matrix).
• Forgetting to check that the rows sum to 1, which is a necessary condition for a valid transition matrix.

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