Current State as a Summary of Relevant History

When Is the Current State Enough?

A multi-state model describes how a process moves among states over time. The central question here is not simply which state labels are clinically recognizable. It is whether knowing the current state gives enough information about the past to characterize future evolution. For example, two people may both be in remission now, but one may have entered remission yesterday while the other has remained there for a year. If their future risks differ because of that duration, the label “remission” alone may not summarize the relevant history.

A state does not need to preserve every detail of the past. It needs to retain the details that matter for the future quantity being modeled. This is a modeling claim, not a consequence of choosing a convenient set of labels. The earlier material on the Markov property describes the formal conditional-independence requirement; here the emphasis is on interpreting what the state must summarize and recognizing when it does not.

The equality concerns conditional distributions, not just whether two groups have the same average outcome in one sample. A stronger way to state it is that, once the current state is known, learning additional information about the observed past does not change the distribution of the future state path. If the model also conditions on measured covariates, those must be included on both sides of the comparison. For a time-inhomogeneous process, the prediction may depend on the current time \(t\) as well as the current state; this does not by itself violate the Markov property.

The relevant history depends on the scientific question. If the future quantity is the chance of failure within 30 days, the state need only summarize history relevant to that 30-day prediction for the stated population and time origin. A state adequate for one prediction need not be adequate for every horizon, outcome, or population.

How Omitted History Creates Different Risks Within a State

Suppose \(S\) is a recorded current-state label, and \(H\) distinguishes past histories that are not represented in that label. For a specified future event \(Y\), the risk among people currently in state \(s\) is an average over the histories represented in that group:

This identity follows by partitioning the group in state \(s\) into its possible history categories and adding the event probability within each category, weighted by the category's share. If history-specific risks differ, then the average risk depends on the mix of histories. Two populations with the same current state can consequently have different state-level risks because they contain different proportions of those histories.

That mixture can be useful as a description of a particular group. But one average for one mixture is not proof that the state is sufficient: another group, or the same group at a later time, may have a different history mix and therefore a different average. A state-only Markov model requires the conditional future distribution not to vary with earlier history once the present state and the model's other conditioning information are held fixed.

A State Can Be Informative Without Being Sufficient

A current-state label may be strongly associated with future outcomes and still omit relevant history. “In remission” can separate patients from those currently ill, for instance, while failing to distinguish remission durations. Likewise, “alive” does not necessarily encode the number or timing of prior episodes of illness. The question is not whether the state is useful, but whether additional past information changes the conditional future distribution.

Sometimes the missing information can be represented by refining the state. For example, instead of one remission state, a model might distinguish remission entered recently from remission of longer duration. If the relevant history is elapsed time since entering a state, a model may instead retain that duration explicitly. These choices change what information the model conditions on; they do not make a given state description sufficient by definition. The refined or augmented description still needs to be evaluated.

Justifying the History-Summary Check

The reason for comparing histories within a current state can be seen directly from conditional probability. Let \(H\) represent a past event or duration category, and let \(Y\) indicate a future event within a fixed horizon. If the current state \(S=s\) is sufficient for this prediction, then adding \(H\) should not change the conditional probability:

for every history category \(h\) with relevant data. If instead these probabilities differ, then either the state is not sufficient for this prediction or the comparison needs to account for other conditioning variables, such as current time or measured covariates. The overall state risk is still the weighted average over histories, since the history categories partition the group:

The second equality is the multiplication rule applied separately within each history category. If all the history-specific risks are equal, changing the weights leaves their average unchanged. If the risks differ, changing the weights can change the average. This is why an apparently stable risk in one sample is not enough to establish sufficiency: the sample may simply have had one particular mix of histories.

Assessing the State Description

In data, the Markov property cannot usually be verified by inspecting a single conditional probability. A practical diagnostic is to compare subsequent outcomes among people in the same current state who differ in specified earlier histories, while aligning or adjusting for the model's time scale and measured covariates. Relevant history candidates should come from the clinical process and study question, rather than being chosen only because a pattern appeared in one dataset.

• Compare within-state histories. Check whether future transition patterns or fixed-horizon risks differ by prior events, state duration, or earlier state sequence among people currently in the same state.
• Check time and covariates. A difference attributed to history may reflect different current times or covariate distributions. Use the intended time origin and conditioning set consistently.
• Check data support. If some histories rarely occur in a state, their risks are imprecise or not estimable. Lack of a detected difference with sparse data is not evidence of equality.
• Check state measurement. Misclassified current states or incomplete event histories can conceal or distort history dependence.
• Validate predictions where possible. Compare predicted and observed outcomes across groups with different history mixes, not only in the population used to fit the model.

These checks diagnose possible inadequacy; they do not prove the Markov property. Failure to detect history dependence may result from limited sample size, coarse measurement, or examining too few histories. Conversely, an apparent difference may be due to confounding by variables that should be included in the conditioning information. Interpretation requires both statistical evidence and a defensible clinical description of what was measured.

Ways to Handle Relevant Past Information

When the current state does not summarize relevant history, several modeling choices are possible. Refining the state can distinguish clinically meaningful history groups, such as remission following a first versus a later episode. Explicitly retaining elapsed time in a state can address duration-related prediction. A model may also condition on measured covariates or use a framework that allows future evolution to depend directly on duration or history. These are alternatives with different assumptions and data requirements; none should be selected solely to make a diagram more detailed.

A coarser state can still be appropriate when the goal is a population-average description for a clearly specified group and its history mix is part of the target. But that average should not be treated as universally applicable to other populations with different histories. In particular, a state-specific average prediction is not automatically a history-independent transition law.

The central limitation is that sufficiency is always relative to a chosen state definition, information set, time scale, and target. Adding history can improve a prediction while making the state space more complex and dividing the available data into smaller groups. A useful model therefore balances an adequate summary of relevant past events against interpretability and the amount of information the data can support.