Checking Coherence Of Predicted Transition Matrices

Why Matrix Coherence Matters

A predicted transition matrix is meant to describe a probability distribution over possible states at a future time. Its entries may each look plausible when inspected separately, yet the collection can still be impossible: a probability could be negative, exceed one, or leave too much or too little probability accounted for across destinations. Checking this internal coherence is a basic audit of the prediction itself. It does not establish that the predictions agree with observed outcomes; that is a different question.

The key is to identify precisely what the entries represent. A full transition matrix describes the state occupied at a specified future time, conditional on the state at the starting time and any specified covariates or history. It is not necessarily a table of probabilities for the first transition out of the origin. This distinction matters when a process can move through intermediate states.

The conditioning information \(z\) may include covariates and, where appropriate, observed history. Predictions in a row must refer to the same origin, time interval, covariate values, and conditioning information. Otherwise, even individually valid probabilities may not form a probability distribution over one common prediction target.

Basic Coherence Conditions

For every origin state \(i\) and destination state \(j\), a valid probability satisfies \(0\le p_{ij}(s,t\mid z)\le 1\). In a complete transition matrix, the destination states are mutually exclusive at time \(t\), and exactly one of them is occupied. Therefore, each row must also satisfy

The row sum is the appropriate total because a row conditions on one particular origin state. The entries in that row give the distribution over future destinations for that origin. Adding entries from different rows has no probability interpretation unless a separate distribution over origin states has been specified.

A model may report only a subset of destinations. In that case, the displayed entries need not sum to one, but the omitted probability must be understood and accounted for. For example, if a row includes two specified destinations but omits an “other state” category, their sum can be less than one. A sum greater than one cannot be explained by omitted destinations. For a transparent audit, it is often useful to display the residual probability explicitly:

where \(\mathcal{J}\) is the set of displayed destinations. This residual is valid only if it is nonnegative and genuinely represents the omitted outcomes, rather than unrecorded probability being treated as if it had a known destination.

Why the Conditions Hold

The conditions follow directly from the probability interpretation, not from a particular estimation method. Conditional on the specified origin and prediction information, the events “in state 1 at time \(t\),” through “in state \(K\) at time \(t\)” are mutually exclusive. Under the exhaustive-state assumption, one of them must occur. Addition of probabilities for disjoint events therefore gives

Each summand is a probability, so it lies between zero and one. This justification also identifies the assumptions behind the check: the entries must share the same conditioning information, and the destination states must be mutually exclusive and exhaustive for the target being predicted. If a state is omitted, or if the entries come from different targets, the row-sum identity may not apply to the displayed values as written.

For a prediction of the first transition out of a state, the corresponding events are instead “no outgoing transition by time \(t\)” and “the first outgoing transition by time \(t\) is to destination \(j\),” for each allowed destination \(j\). With a complete list of competing destinations, their probabilities plus the probability of remaining in the origin sum to one. This is the appropriate accounting for competing outgoing transitions. It is not automatically the same as a row of state-occupation probabilities: after leaving the origin, a process may enter an intermediate state and then move again before time \(t\).

Coherence Across Time Intervals

A matrix may pass every row-sum check and still conflict with the model’s assumptions about transitions over time. Under a Markov model, the distribution after two consecutive intervals must agree with the distribution obtained by composing the interval-specific transition matrices. The Markov assumption here means that, conditional on the state at an intermediate time and the relevant modeled covariates, the future distribution does not additionally depend on earlier history.

when these matrices describe the same process and use compatible covariate and time specifications. The multiplication sums over every possible intermediate state: the probability of ending in \(j\) from \(i\) is the sum, over intermediate states \(k\), of the probability of reaching \(k\) and then reaching \(j\). This is a stronger check than row sums, and it depends on the Markov specification. It should not be imposed blindly when the prediction target or model does not have that property.

Practical Audit and Failure Modes

A useful audit proceeds from the definition of the prediction target to the numerical checks. It should be performed for every origin row and relevant prediction time, not just on an overall average of the matrix.

Small discrepancies may arise from rounding displayed values. For instance, a row shown as 0.333, 0.333, and 0.333 sums to 0.999; unrounded values may sum to one. The audit should therefore use unrounded predictions where available and report the display precision. Rounding cannot plausibly explain substantial discrepancies, negative probabilities, or values above one.

Structural constraints provide further checks. If a transition is impossible by definition, its probability should be zero. If a state is absorbing under the specified model, its row should assign probability one to remaining in that state at every horizon. These checks depend on the model’s state definitions and assumptions; they are not universal properties of every state with a particular label.

Coherence is necessary, not sufficient. A matrix can satisfy every numerical constraint and still be poorly estimated or systematically disagree with observed outcomes. Conversely, estimating each destination probability with a separate model can produce values that look plausible individually but fail to form a joint distribution. Joint multinomial or competing-transition approaches can enforce shared probability accounting by construction; constructing a transition matrix from a coherent transition model is another option. Independently fitted probabilities may require a revised model or a carefully justified reconciliation step. Mechanical truncation of negative values, clipping values above one, or normalizing rows changes predictions and does not by itself establish a valid statistical model.