Conditioning on the Current State in Multi-State Prediction

Why Condition on the State Occupied Now?

At a landmark time, people in the same multi-state process may occupy different states. A forecast for someone currently in state A need not resemble a forecast for someone currently in state B: their available transitions and risks of reaching later states may differ. The practical task is therefore to calculate and report predictions separately by current state, rather than treating the landmark population as if everyone began from one common state.

The key operation is simple once the estimand is clear: for a specified landmark time and prediction horizon, use the transition probabilities conditional on the state occupied at the landmark. This tutorial focuses on that conditioning and on what happens when state-specific forecasts are combined. It does not require that everyone have the same observed history before the landmark; it does require care about whether the current state is sufficient for the prediction model being used.

Definition: Let $X(t)$ be the state occupied at time $t$, and let the landmark time be $s$. For a person in state $h$ at $s$, the state-occupation probability for state $j$ at a later time $t$ is $$ p_{hj}(s,t)=P\{X(t)=j\mid X(s)=h\},\qquad t\ge s. $$ The state-specific prediction is the vector $p_h(s,t)=(p_{h1}(s,t),\ldots,p_{hK}(s,t))$, where $K$ is the number of states. It describes the distribution of states at $t$ conditional on occupying $h$ at $s$.

This is a conditional probability for a defined landmark population, not an individual guarantee and not a causal effect of being in state $h$. If the landmark cohort is restricted to people eligible and observed at $s$, that restriction is part of the target population. The prediction horizon $t-s$, the state definitions, and the landmark time should be specified so that the conditioning event and intended use are unambiguous.

Reading the Appropriate Row of a Transition Matrix

Write $P(s,t)$ for the transition probability matrix from $s$ to $t$. We use the convention that rows index the state at $s$ and columns index the state at $t$: its $(h,j)$ entry is $p_{hj}(s,t)$. Thus, the state-specific forecast for people currently in state $h$ is row $h$ of $P(s,t)$. If the current cohort is in several states, each state has its own conditional row.

Each row is a probability distribution: its entries are nonnegative and sum to one, provided the state system accounts for every possible state at the prediction time. Absorbing states, such as death in a model where death is absorbing, remain in the matrix; their rows assign probability one to remaining in that state. The state-occupation estimand and transition-matrix interpretation are the basis for this calculation.

Worked Example: Separate Predictions for Two Current States

Suppose a hypothetical three-state process has states A, B, and D, where D is absorbing. For a landmark at $s$ and a horizon $t$, suppose the transition probability matrix is $$ P(s,t)= \begin{pmatrix} 0.60&0.25&0.15\\ 0.10&0.65&0.25\\ 0&0&1 \end{pmatrix}. $$ The rows correspond to current states A, B, and D, in that order, and the columns to states at $t$ in the same order.

For a person currently in A, select the first row: $$ p_A(s,t)=(0.60,0.25,0.15). $$ Thus the predicted probabilities of occupying A, B, and D at $t$, conditional on A at $s$, are 0.600, 0.250, and 0.150. For a person currently in B, select the second row: $$ p_B(s,t)=(0.10,0.65,0.25). $$ The corresponding predicted probabilities are 0.100, 0.650, and 0.250. The row sums are $0.60+0.25+0.15=1.00$ and $0.10+0.65+0.25=1.00$.

If, hypothetically, 120 eligible people occupy A at the landmark, the expected numbers in A, B, and D at $t$ under this prediction are $120(0.60)=72$, $120(0.25)=30$, and $120(0.15)=18$. For 80 people in B, they are $80(0.10)=8$, $80(0.65)=52$, and $80(0.25)=20$. These are model-based expected counts, not claims about the exact future outcomes of those particular people. The conditional forecasts differ because the starting states differ.

Prediction Across More Than One Interval

A prediction may be constructed over multiple intervals. If $s<u<t$, the transition matrix from $s$ to $t$ is obtained by multiplying the matrix from $s$ to $u$ by the matrix from $u$ to $t$, in that order. This multiplication sums over the possible intermediate states at $u$. The Markov assumption is what allows the second interval's transition probabilities to depend on the state at $u$, rather than additionally on the earlier state at $s$.

Assumption for the matrix composition: The process is Markov at the times being used: conditional on $X(u)$, the distribution of the future state $X(t)$ does not additionally depend on the state history before $u$. Transition probabilities may vary with calendar time; the composition rule does not by itself require time-homogeneous transitions.

Worked Example: Propagating a Forecast Through an Intermediate Time

For the same hypothetical states A, B, and D, suppose the matrices for two intervals are $$ P(s,u)= \begin{pmatrix} 0.80&0.20&0\\ 0&0.70&0.30\\ 0&0&1 \end{pmatrix}, \qquad P(u,t)= \begin{pmatrix} 0.60&0.30&0.10\\ 0.10&0.60&0.30\\ 0&0&1 \end{pmatrix}. $$ For the forecast conditional on A at $s$, the row at $s$ is $(0.80,0.20,0)$. Multiplying by the second matrix gives $$ (0.80,0.20,0)P(u,t) =0.80(0.60,0.30,0.10)+0.20(0.10,0.60,0.30) =(0.50,0.36,0.14). $$ So, conditional on A at $s$, the predicted probabilities at $t$ are 0.500 for A, 0.360 for B, and 0.140 for D.

For a person in B at $s$, the row at the first interval is $(0,0.70,0.30)$. Therefore, $$ (0,0.70,0.30)P(u,t) =0.70(0.10,0.60,0.30)+0.30(0,0,1) =(0.07,0.42,0.51). $$ The predictions conditional on B at $s$ are 0.070, 0.420, and 0.510 for A, B, and D. Each resulting vector sums to one. For example, $0.07+0.42+0.51=1.00$. The forecasts differ by starting state even though they share the same transition matrices.

Why Matrix Composition Works

To obtain the probability of state $j$ at $t$ conditional on state $h$ at $s$, partition according to the state $k$ occupied at the intermediate time $u$. The law of total probability gives a sum over all such states. Under the Markov assumption, the conditional probability of the later state given the intermediate state no longer depends on the state at $s$. Hence

\begin{aligned} P\{X(t)=j\mid X(s)=h\} &=\sum_{k=1}^{K}P\{X(t)=j\mid X(u)=k,X(s)=h\} P\{X(u)=k\mid X(s)=h\}\\ &=\sum_{k=1}^{K}P\{X(t)=j\mid X(u)=k\} P\{X(u)=k\mid X(s)=h\}. \end{aligned}

The final expression is the $(h,j)$ entry of $P(s,u)P(u,t)$. This derivation explains both the order of multiplication and why the intermediate states must be summed over. It also identifies the key condition: if earlier history still predicts the future after conditioning on the current state, this composition of state-only transition matrices need not describe the desired conditional prediction.

Combining State-Specific Predictions for a Target Cohort

Separate conditional forecasts answer a different question from a forecast for a cohort containing several current states. To obtain a prediction for a mixed cohort, weight the state-specific vectors by the cohort's proportions in the corresponding landmark states. Equal weighting is appropriate only when the target cohort has equal proportions in those states. A pooled forecast based on another state composition targets another population.

Worked Example: Weighting Predictions for a Mixed Landmark Cohort

Return to the matrix in the first example. Suppose a hypothetical target cohort of 100 people comprises 30 people in A and 70 in B at the landmark. The state proportions are therefore 0.30 and 0.70. Its forecast for state A at $t$ is $$ 0.30(0.60)+0.70(0.10)=0.18+0.07=0.25. $$ For state B at $t$, it is $$ 0.30(0.25)+0.70(0.65)=0.075+0.455=0.53. $$ For state D at $t$, it is $$ 0.30(0.15)+0.70(0.25)=0.045+0.175=0.22. $$ The mixed-cohort prediction is thus $(0.25,0.53,0.22)$, which sums to one. In expected counts, this corresponds to 25 in A, 53 in B, and 22 in D out of 100.

This mixture does not replace the conditional forecasts $(0.60,0.25,0.15)$ for A and $(0.10,0.65,0.25)$ for B. It answers a different question: what is predicted for a target cohort with the stated 30:70 distribution across current states?

Conditions, Diagnostics, and Alternatives

A valid interpretation starts with a clearly specified prediction target and a suitable transition model. In practice, check the following conditions and potential failure modes:

State and landmark are observed consistently. State definitions should be mutually exclusive and sufficiently complete for the intended process. The landmark population must be defined, including eligibility and observation requirements at $s$.
The current state is adequate for the model. State-only transition probabilities require the Markov property at the prediction times. If people in the same current state have systematically different future transition patterns because of earlier history, the state-only prediction may be inadequate. A row sum of one does not establish the Markov assumption.
Each starting state has support. If few or no people occupy a state at the landmark, its conditional forecast may be unstable or not estimable from the data. A numerical output from a model does not create empirical support where none exists.
Observation and censoring allow estimation. When transition probabilities are estimated from follow-up data, state changes must be recorded accurately, and censoring assumptions must be appropriate for the estimation method and risk sets. In particular, informative loss to follow-up can distort estimated transitions if not adequately addressed.
Check each row, not only the mixture. Probabilities should be between zero and one, and each complete state distribution should sum to one up to rounding. A plausible pooled distribution can conceal an incorrectly selected row or an error in a state-specific calculation.

The main alternative summaries answer distinct questions. A single pooled state distribution averages over the landmark cohort's state composition and should not be interpreted as the forecast for every current state. A forecast from a fixed initial state answers the question for that specified starting state only. A transition-model approach can produce state-specific rows directly, while a nonparametric transition-matrix estimate uses observed transitions and risk sets under its assumptions. These approaches may differ in estimation and modeling assumptions, but the conditioning event must still match the question.

State-specific prediction is useful when clinical decisions or communication depend on where a person is in the process now. Its limitations matter: a conditional population forecast is not necessarily a reliable individualized forecast, and it does not automatically account for information beyond the current state. It can also be imprecise for rare starting states or long prediction horizons. These limitations motivate checking the target population, model assumptions, and data support before comparing or applying the separate rows.

Common Mistake

Pitfall: “The pooled cohort has a 22% predicted probability of D, so a person currently in A has a 22% probability of D.” This is wrong when the pooled cohort mixes people in different current states. In the weighted example, 22% was the forecast for a cohort with 30% in A and 70% in B. The corresponding conditional probability for someone in A was 15%, and for someone in B it was 25%. Always identify the conditioning state before interpreting a probability; use a weighted average only for the target cohort whose state composition supplies those weights.

Check Your Understanding

Use the definitions and calculations above to answer these questions. Brief solutions follow each question.

A transition matrix has rows indexed by landmark state and columns by later state. Which row gives the forecast for a person in state 2 at the landmark?
Solution: Row 2, because its entries are conditional probabilities given state 2 at the landmark.
For a current-state row $(0.40,0.35,0.25)$, what is the predicted probability of occupying the third state, and do the entries form a complete distribution?
Solution: The probability of the third state is 0.25. The entries sum to $0.40+0.35+0.25=1.00$, so they form a complete probability distribution.
A cohort contains 40% of people in A and 60% in B. Their predicted probabilities of state D are 0.10 and 0.30, respectively. What is the mixed-cohort prediction for D?
Solution: Weight by the cohort proportions: $0.40(0.10)+0.60(0.30)=0.04+0.18=0.22$. The mixed-cohort probability is 0.22.
Why does matrix multiplication combine two successive transition intervals?
Solution: It sums over each possible intermediate state at the end of the first interval, multiplying the probability of reaching that state by the probability of the later state conditional on it. The Markov assumption permits the second probability to depend on the intermediate state rather than earlier state history.
A row has entries $(0.55,0.30,0.15)$, but the analyst reports the first entry as the forecast for people in a different landmark state. What is the error?
Solution: The analyst selected the wrong row. The first entry is the probability of the first later state conditional on the state indexing that row, not a general probability for all landmark cohorts.