Transition Probability Matrix

Definition

The Transition Probability Matrix (TPM) is the full set of all transition probabilities $P_{ij}$ in matrix form:

$$\mathbf{P}=\Vert \begin{array}{cccc}{P}_{00}& P_{01}& P_{02}& \cdots \\ P_{10}& P_{11}& P_{12}& \cdots \\ ⋮& ⋮& ⋮& \\ P_{i0}& P_{i1}& P_{i2}& \cdots \\ ⋮& ⋮& ⋮& \end{array}\Vert$$

Where $P_{ij}$ is the transition probability from state $i$ to state $j$.

Interpretation

• $P_{12}$ — probability of transitioning from state 1 to state 2 in one step
• $P_{34}$ — transition probability from state 3 to 4
• $P_{55}$ — transition probability from state 5 to 5

Properties

Each row sums to 1: ${\sum }_j{P}_{ij}=1$ for all $i$

Procedure: Validating TPM

1. All entries $P_{ij}\ge 0$
2. Each row sums to 1: ${\sum }_j{P}_{ij}=1$ for all $i$

Related

• Transition Probability
• n-step Transition Matrix
• Regular TPM