From an Initial Distribution to a Future Distribution
A multiple-state model can describe a person’s status as, for example, active, disabled, or dead. Often, the starting state is not known to be the same for every person in a group. Instead, the model begins with a probability distribution across the possible starting states. The practical question is then: what is the distribution across states after a specified period?
The calculation is a matrix multiplication. Its result gives the probability of being in each state at the future time, without requiring that everyone began in the same state. The state order must be consistent in both the initial distribution and the transition probability matrix.
The entries of the transition matrix follow the convention that \(p_{ij}(0,t)\) is the probability of being in state \(j\) at time \(t\), conditional on being in state \(i\) at time 0. Thus, each row describes the possible destinations from one starting state. If the states are exhaustive, every row sums to 1, and all entries are between 0 and 1.
The calculation applies when the initial state probabilities and the transition probabilities refer to the same population, time interval, state definitions, and state ordering. The transition matrix may be obtained from a model or supplied directly. No assumption of constant transition intensities is needed if the matrix for the interval is already given. If transition probabilities are supplied for a different interval, they cannot be used without first obtaining the appropriate matrix for the interval of interest.
Why the Multiplication Works
To find the probability of occupying a particular destination state \(j\) at time \(t\), partition the possible histories according to the state \(i\) occupied at time 0. These starting-state events are disjoint and cover all possibilities. For each \(i\), multiply the probability of starting in \(i\) by the conditional probability of ending in \(j\) given that start.
The final expression is exactly component \(j\) of the row-vector product \(\boldsymbol{\pi}(0)P(0,t)\). This also explains the orientation: the \(i\)th initial probability weights row \(i\), and the resulting component for state \(j\) adds the contributions to \(j\) from every possible starting state.
Write the same ordered list for the initial vector and for the rows and columns of the transition matrix.
The initial probabilities should sum to 1. Each row of a complete transition probability matrix should also sum to 1.
Compute the row vector \(\boldsymbol{\pi}(0)P(0,t)\). Each output component is a probability for one destination state.
Attach the state labels to the output components and check that they sum to 1, allowing only for stated rounding.
Worked Calculations
Worked Example: A Cohort Initially in Three Possible States
A hypothetical group has three states, ordered as Active (A), Disabled (D), and Dead (X). At time 0, the probabilities of these states are \(0.80\), \(0.15\), and \(0.05\), respectively. For the next year, the transition probability matrix in this state order is
Find the distribution of states after one year. The method is to multiply the initial row vector by the transition matrix. The rows sum to 1: \(0.75+0.20+0.05=1\), \(0.10+0.70+0.20=1\), and \(0+0+1=1\).
After one year, the probabilities of being Active, Disabled, and Dead are \(0.615\), \(0.265\), and \(0.120\). As a check, \(0.615+0.265+0.120=1.000\). The calculation includes, for example, people who started Disabled and became Active in the first component, through the term \(0.15(0.10)\). A common mistake is to use only the first row because the Active state is listed first; that would incorrectly assume everyone started Active.
Worked Example: Finding a State Probability and an Event Probability
A hypothetical long-term care model has states Healthy (H), Receiving Care (C), and Dead (D), in that order. At the start of a two-year interval, the state distribution is \((0.60,0.30,0.10)\). The transition probability matrix for the entire two-year interval is
Find the probability of receiving care at the end of two years and the probability of being alive at that time. Since the supplied matrix already covers two years, use it once for this interval.
The probability of receiving care at the end of the interval is the second component, \(0.315\). Being alive means being Healthy or Receiving Care, so the probability of being alive is \(0.51+0.315=0.825\). Equivalently, \(1-0.175=0.825\). The output sums to \(0.51+0.315+0.175=1\). The common error avoided here is treating “ever received care during the interval” as the question: this matrix multiplication gives endpoint states, not probabilities of having visited a state at any time.
Worked Example: Translating Probabilities into Expected Counts
A hypothetical insurer models 400 policyholders in the ordered states Active (A), Disabled (D), and Dead (X). At the start of a six-month interval, 280 are Active, 80 are Disabled, and 40 are Dead. The transition probability matrix for that six-month interval is
Find the future state distribution and the expected number of policyholders in each state. First convert the initial counts to probabilities: \(280/400=0.70\), \(80/400=0.20\), and \(40/400=0.10\). Propagate this row vector through the matrix:
The expected counts are found by multiplying each probability by 400: Active \(=400(0.590)=236\), Disabled \(=400(0.235)=94\), and Dead \(=400(0.175)=70\). The counts sum to \(236+94+70=400\), matching the original group size. The method avoids multiplying counts by a matrix and then interpreting the result as probabilities: counts can be propagated directly, but their outputs are counts, not proportions. Here the probability calculation makes the units explicit.
Interpreting the Result and Avoiding Errors
Each component of \(\boldsymbol{\pi}(t)\) is a probability of occupying a state at the specified endpoint. It is not, by itself, a probability of having ever entered that state, nor does it give an individual’s actual future state. For a group of \(N\) people with the same initial distribution and transition model, \(N\pi_j(t)\) is the expected number in state \(j\). It need not be an integer when the expected count is not whole.
Before relying on a result, check the state order, the time interval, and the direction of multiplication. With the row-vector convention used here, the initial distribution is on the left and the transition matrix is on the right. A column-vector convention can also be used in other settings, but it requires a corresponding change in matrix orientation; mixing the conventions gives an incorrect calculation.
| Check | What to verify |
|---|---|
| State ordering | The initial vector and transition matrix use the same labels in the same order. |
| Initial distribution | Its components are probabilities and sum to 1. |
| Transition matrix | Each row lists destinations conditional on one starting state and sums to 1. |
| Time interval | The matrix covers the interval from the initial time to the requested future time. |
| Output | Components are endpoint probabilities, are nonnegative, and sum to 1. |
These checks are especially useful when a state is absorbing, such as Dead in the examples: its row has probability 1 of remaining there. That row contributes to the future distribution according to the initial probability of already being in that state. It does not mean that everyone ultimately moves to the absorbing state during the interval.
Check Your Understanding
For each question, use the stated state order and report the requested endpoint probabilities.
- A model has states A, B, and X, where X is absorbing. The initial distribution is \((0.40,0.50,0.10)\), and the transition matrix is \(\begin{pmatrix}0.60&0.30&0.10\\0.20&0.60&0.20\\0&0&1\end{pmatrix}\). Find the future distribution. (Answer: \((0.34,\ 0.42,\ 0.24)\))
- In state order H, C, D, the initial distribution is \((0.25,0.50,0.25)\), and the interval transition matrix is \(\begin{pmatrix}0.80&0.15&0.05\\0.10&0.70&0.20\\0&0&1\end{pmatrix}\). Find the future distribution. (Answer: \((0.25,\ 0.3875,\ 0.3625)\))
- A cohort of 200 is initially distributed across A, B, and X in proportions \((0.60,0.40,0)\). The transition matrix is \(\begin{pmatrix}0.90&0.10&0\\0.25&0.65&0.10\\0&0&1\end{pmatrix}\). Find the expected future count in each state. (Answer: A 128, B 64, X 8)
- For states 1 and 2, the initial distribution is \((0.30,0.70)\) and the transition matrix for the requested interval is \(\begin{pmatrix}0.85&0.15\\0.40&0.60\end{pmatrix}\). Find the future distribution and verify its total. (Answer: \((0.535,\ 0.465)\); total \(1.000\))