Why State-Occupation Probabilities Sum to One

Why Probability Conservation Matters

A state-occupation prediction answers a question about where a person is at a specified future time: for example, in an initial disease state, in a progressed state, or dead. If those categories cover every possible status and cannot occur simultaneously, the probabilities assigned to them must add to one. This is a basic consistency condition, not evidence by itself that the prediction model is well specified or calibrated.

The condition applies at each prediction horizon and for each fixed prediction target. In landmark prediction, that target is usually defined by the person’s observed history and covariates at the landmark. Predictions at different horizons can have different probability distributions, but each distribution must account for the full state space at its own horizon.

Definition: Let \(S=\{1,\ldots,K\}\) be a finite collection of states. It is mutually exclusive if a person can occupy at most one of these states at a given time, and exhaustive if the person must occupy one of them. For a person with landmark information \(H_L\) and covariates \(X\), define the state-occupation probability for state \(j\) at horizon \(u\geq 0\) as \(p_j(u\mid H_L,X)=P\{Z(L+u)=j\mid H_L,X\}\). If \(S\) is mutually exclusive and exhaustive for the target, then \(\sum_{j=1}^{K}p_j(u\mid H_L,X)=1\) at every horizon \(u\).

The requirement concerns the probabilities for one well-defined target and horizon. It does not say that the probability of occupying a particular state must stay constant over time. Nor does it require separately reported confidence-interval endpoints for different states to sum to one: such intervals are marginal summaries, not a joint state-probability vector.

A Direct Justification

For a fixed horizon, let \(I_j\) equal 1 if the person is in state \(j\) and 0 otherwise. Exactly one indicator equals 1, so \(\sum_{j=1}^{K} I_j=1\). Taking conditional expectations given the landmark information and covariates gives the result:

\sum_{j=1}^{K}p_j(u\mid H_L,X) =\sum_{j=1}^{K}E\{I_j\mid H_L,X\} =E\left\{\sum_{j=1}^{K}I_j\mid H_L,X\right\} =E(1\mid H_L,X) =1.

This argument uses the definition of the target states, not a particular estimation method. It holds whether the probabilities come from a transition-based model, a direct model for state occupation, or a correctly defined population distribution. The crucial conditions are that the states are exhaustive and mutually exclusive, and that all probabilities refer to the same horizon and conditioning information.

Transition Matrices Preserve the Sum

A transition-based calculation gives a second way to see the constraint. Suppose the state probabilities at one time are represented by a row vector \(\boldsymbol{\pi}\), and the transition matrix \(P\) gives the conditional probability of moving from each current state (row) to each next state (column). For a complete state space, every entry of \(P\) is nonnegative and each row sums to one. A vector of state probabilities propagated by this matrix is \(\boldsymbol{\pi}P\).

Result: If the entries of \(\boldsymbol{\pi}\) are nonnegative and sum to one, and \(P\) is a row-stochastic transition matrix (nonnegative entries with every row summing to one), then \(\boldsymbol{\pi}P\) also has nonnegative entries summing to one.

To verify the claim, write the \(j\)th entry of the propagated vector as \((\boldsymbol{\pi}P)_j=\sum_i\pi_iP_{ij}\). Summing over destination states and rearranging the finite sums gives:

\sum_j(\boldsymbol{\pi}P)_j =\sum_j\sum_i\pi_iP_{ij} =\sum_i\pi_i\left(\sum_jP_{ij}\right) =\sum_i\pi_i =1.

The same argument applies at successive steps: if \(P_1,\ldots,P_m\) are row-stochastic transition matrices, then \(\boldsymbol{\pi}P_1\cdots P_m\) sums to one. This is a bookkeeping identity for complete probability distributions. It does not establish that the transition probabilities are estimated well, that the chosen state structure is appropriate, or that the model’s assumptions hold.

Worked Example: Checking a Landmark State Distribution

Suppose a hypothetical model predicts the following probabilities one year after a landmark: 0.62 in the initial state, 0.23 in a progressed state, and 0.15 in an absorbing event state. Assume these three states are defined as mutually exclusive and exhaustive for the target population at one year.

The sum is \(0.62+0.23+0.15=1.00\). Each value is between 0 and 1, and their total is one, so this vector satisfies the necessary probability constraint. It says that, for the specified landmark target, the predicted one-year state distribution assigns 62% probability to the initial state, 23% to the progressed state, and 15% to the event state.

This check is necessary but not sufficient. The arithmetic does not show that the model is calibrated, that the chosen states fully represent the scientific outcome, or that the predictions are accurate. Those questions require other checks and substantive justification.

Worked Example: Propagating Probabilities Through a Transition Matrix

Consider a hypothetical discrete-time model with states A, B, and D, where D is absorbing. At each step, let the transition matrix be

P= \begin{pmatrix} 0.7&0.2&0.1\\ 0.1&0.6&0.3\\ 0&0&1 \end{pmatrix}.

The rows correspond to the current state and the columns to the next state. The row sums are \(0.7+0.2+0.1=1\), \(0.1+0.6+0.3=1\), and \(0+0+1=1\). Thus, under the model’s complete state space, every row is a probability distribution. Suppose everyone in the prediction target is in A at the landmark, so \(\boldsymbol{\pi}_0=(1,0,0)\).

After one step, the distribution is \(\boldsymbol{\pi}_1=\boldsymbol{\pi}_0P=(0.7,0.2,0.1)\), which sums to \(0.7+0.2+0.1=1\). After two steps, multiply the one-step distribution by \(P\). The probability of A is \(0.7(0.7)+0.2(0.1)+0.1(0)=0.51\). The probability of B is \(0.7(0.2)+0.2(0.6)+0.1(0)=0.26\). The probability of D is \(0.7(0.1)+0.2(0.3)+0.1(1)=0.07+0.06+0.10=0.23\).

Therefore, \(\boldsymbol{\pi}_2=(0.51,0.26,0.23)\), and \(0.51+0.26+0.23=1.00\). The contribution to D includes people already in D after the first step; because D is absorbing, their probability \(0.10\) remains in D. The calculation illustrates conservation of total probability across state transitions.

What a Violation Can Reveal

A reported state-probability vector that materially fails to sum to one cannot be a valid distribution over one mutually exclusive, exhaustive state space for one target and horizon. The discrepancy is a signal to inspect the definitions and calculation before interpreting the values. Common causes include:

An omitted state. For example, predictions for several non-event states may exclude death, even though death is possible by the horizon. The listed categories then do not exhaust the target’s possible statuses.
Overlapping categories. “Ever progressed by the horizon” and “currently in the progressed state” are not necessarily mutually exclusive or equivalent outcomes. Probabilities for overlapping events need not sum to one.
Different targets or horizons. Values conditioned on different landmark histories, covariate profiles, or time horizons do not generally form one distribution and should not be added as if they did.
Unconstrained separate models. Fitting a separate binary model for each state can produce predictions that do not add to one, because the models do not enforce a shared probability distribution.
Implementation or numerical error. Incorrect matrix orientation, an omitted transition, or a row of transition probabilities that does not sum to one can break a transition-based calculation. Small discrepancies can also arise from rounding displayed values.

Censoring requires careful interpretation. In a usual prediction of the underlying state at a future time, loss to follow-up is an observation process, not automatically an additional disease state. The state probabilities for the target population still cover the underlying states. By contrast, if the prediction target explicitly includes observation status, then that status must be defined as part of the state space before applying the sum-to-one condition.

Worked Example: Interpreting an Inconsistent Output

A hypothetical prediction program reports, at the same two-year horizon and for the same landmark target, probabilities 0.58 for “state A,” 0.31 for “state B,” and 0.19 for “state D.” The intended definitions say these three states are mutually exclusive and exhaustive.

The total is \(0.58+0.31+0.19=1.08\), which exceeds one by 0.08. These cannot all be valid state-occupation probabilities with the stated definitions. The excess is too large to explain by displaying values to two decimal places alone: ordinary rounding of three values to two decimal places can change their sum by at most about 0.015.

A possible diagnostic is to inspect whether the three values came from separate binary models rather than a joint state model. Dividing each value by 1.08 would produce approximately \(0.5370\), \(0.2870\), and \(0.1759\), which sum to one apart from rounding. But that arithmetic rescaling does not show that the adjusted values estimate the intended state probabilities. It changes all three predictions and may conceal overlapping categories, inconsistent targets, or a coding error. The appropriate response is to check the definitions and model construction, then use a method whose output represents a joint distribution over the specified states.

Diagnostics, Alternatives, and Limits

A practical check should be performed on the full-precision probabilities, not only on values rounded for a plot or table. For every prediction target and horizon, inspect whether each component is between 0 and 1 and whether the components sum to one within a tolerance appropriate to the numerical computation. For transition-based predictions, also inspect the transition matrix: entries should be nonnegative and each row should sum to one. Confirm that matrix rows and columns follow the stated current-state and destination-state convention.

Small discrepancies in displayed output can result from rounding. For instance, three full-precision values that sum to one might be displayed as 0.33, 0.33, and 0.34; conversely, displayed values may sum to 0.99 or 1.01 even when their unrounded values sum to one. This is different from a substantial discrepancy in the full-precision calculation. Do not silently adjust displayed values unless the reporting method clearly states that rounding is responsible.

Transition-based models enforce the constraint when their initial distribution and transition matrices are valid probability distributions over a complete state space. Direct models for a multicategory state outcome can also enforce it by construction. In contrast, separate binary predictions for each state may not. These approaches have different modeling assumptions and uses; the sum-to-one check does not determine which approach is preferable for a particular scientific question. It checks internal coherence, not model fit.

The condition also has a useful interpretation in competing risks. If the state space includes being event-free at the horizon and every possible absorbing event type, the event-free probability plus the probabilities of occupying the event states sums to one. Omitting a competing event type, or reporting only event probabilities without the event-free state, changes what is being summed; the resulting quantities need not total one.

Finally, a valid point prediction does not guarantee valid uncertainty quantification. Separately calculated interval bounds can fail to sum to one without contradicting the pointwise identity. Nor does a sum of one establish calibration: a model can allocate all probability across the states and still systematically assign too much probability to one state and too little to another. Probability conservation is a necessary structural check, not a complete evaluation.

Key takeaway: At a fixed horizon and for a fixed target, mutually exclusive and exhaustive state-occupation probabilities must sum to one. When a reported vector does not, investigate the state definitions, target, horizon, model construction, and computation before interpreting or rescaling it.

Check Your Understanding

Answer each question using the probability-conservation condition and its assumptions.

A prediction gives probabilities 0.54, 0.29, and 0.17 for three mutually exclusive and exhaustive states at one horizon. Does it satisfy the sum constraint?
Why does multiplying a probability vector by a row-stochastic transition matrix preserve its sum?
Three separately fitted binary models give state probabilities 0.45, 0.40, and 0.30. What should be checked before using these as one state distribution?
Can the lower endpoints of three separate 95% confidence intervals for state probabilities be required to sum to one? Explain.

Solutions. (1) Yes: \(0.54+0.29+0.17=1.00\). This verifies the sum constraint, assuming the target and state definitions are as stated. (2) Each row of the transition matrix sums to one, so summing the propagated vector gives the sum of the original vector multiplied by those row sums; the total remains one. (3) Their sum is \(0.45+0.40+0.30=1.15\). Check that all predictions refer to the same target and horizon, that categories are exclusive and exhaustive, and whether separate models failed to enforce a joint distribution. (4) No. Marginal interval endpoints are not themselves a joint state-probability vector; the sum-to-one identity applies to the point probabilities for the same target and horizon, not to separately constructed interval bounds.