Unobserved Transitions Between Panel Visits

Why Unobserved Transitions Matter

At a panel visit, a subject’s state is observed; between visits, it may not be. If the state is \(A\) at both ends of an interval, that observation does not by itself establish that the subject remained in \(A\). The subject may have left \(A\) and returned before the next visit. A likelihood that counts only paths consistent with “no change” can therefore omit possible histories and misrepresent the probability of the observed data.

The key calculation is to add the probabilities of all latent state sequences compatible with the observed endpoints. The transition-matrix likelihood for panel data supplies the endpoint probability; splitting an interval at an unobserved time makes the hidden-state summation explicit. This tutorial focuses on that summation, not on estimating a particular model’s transition parameters.

The state process must be Markov at the split times used for the calculation: conditional on the current state (and any modeled covariates or history), its future transition distribution cannot depend further on its earlier path. The state space and transition probabilities must also correctly represent the allowed transitions. If the visit schedule is informative about the unobserved path, its probability may need to be modeled too; the calculations below treat the schedule as fixed or noninformative for the state process.

Summing Over Hidden States

Suppose an observation interval from \(s\) to \(t\) is split at an unobserved time \(u\). Write \(P^{(1)}=P(s,u)\) and \(P^{(2)}=P(u,t)\). The state at \(u\) could be any state \(k\) in \(S\). Conditional on that state, the Markov property gives the probability of the two-part path as \(P^{(1)}_{ik}P^{(2)}_{kj}\). Because the intermediate state is not observed, add these mutually exclusive possibilities:

This is matrix multiplication. With more split times, sum over every sequence of hidden states, multiplying the transition probabilities along that sequence:

A term in this sum is the probability associated with one state sequence on the chosen time grid. Importantly, a matrix entry for a subinterval may itself aggregate paths within that subinterval. Thus, the grid describes which intermediate states are explicitly summed over; it does not necessarily enumerate every transition time.

Return Paths in a Larger State Space

With more than two states, several distinct hidden states may provide routes back to the observed state. Each compatible route contributes its own product of transition probabilities. Keeping these terms separate during calculation can make it easier to verify that no allowed path has been omitted.

From Interval Contributions to a Subject Likelihood

For observations \(x_0,x_1,\ldots,x_m\) at times \(t_0<t_1<\cdots<t_m\), the likelihood contribution for one subject under a Markov model is the product of the transition probabilities across consecutive visits. At each interval, its endpoint probability already sums over compatible hidden paths in that interval. Conditional independence across subjects then gives the full likelihood as the product of their subject-level contributions.

Here \(n\) indexes subjects, \(r\) indexes their observed intervals, and \(\theta\) denotes model parameters. This product uses observed states only. It does not require selecting a single hidden path for each subject. If a model instead includes a latent state at a split time, summing over that state recovers the same endpoint contribution.

Derivation, Checks, and Failure Modes

The hidden-state sum follows directly from conditioning on the state at the split. For the event \(X(s)=i\) and \(X(t)=j\), the events \(\{X(u)=k\}\), \(k\in S\), partition all possible states at \(u\). By the Markov property, for each such \(k\),

Adding over the disjoint possibilities yields

Repeated conditioning gives the corresponding sum over state sequences for any number of splits. This derivation depends on having a state process for which the Markov property holds at the split times; an omitted variable that carries relevant information about the future can invalidate the factorization.

Useful checks include verifying that every transition matrix has nonnegative entries and rows summing to 1, and that each endpoint probability lies between 0 and 1. If all endpoint probabilities are computed, the resulting transition matrix should also have rows summing to 1. For a same-state endpoint, check explicitly whether the model permits departures and returns. If it does, confirm that all allowed hidden-state routes have been included.

• Dropping return paths. Keeping only the “stay in the state” term confuses equal observed endpoints with an unchanged latent history. Sum over all compatible hidden states instead.
• Using a forbidden route. A path through a state that cannot be reached under the specified model must have probability zero, not an invented positive contribution. Check the model’s transition structure.
• Multiplying instead of summing over alternatives. Hidden states at a split are mutually exclusive possibilities, so their path probabilities are added. Probabilities across consecutive observed intervals are multiplied.
• Assuming the state alone is sufficient. If the future depends on prior history beyond the current state, the Markov factorization may fail unless the state definition is expanded or the dependence is modeled.

An alternative is to use the interval transition matrix directly, without explicitly listing hidden paths; its relevant entry is already the required endpoint probability. Explicit path summation is useful for understanding or checking that entry. Choosing a single “most likely” path is not equivalent to summing: it discards the probability mass of other compatible paths. In irreversible models, some return routes are impossible, but same-state endpoint probabilities still need to reflect every route the model does allow. The main limitation is that observed endpoints alone generally do not reveal the path or identify every transition probability; the likelihood calculation accounts for uncertainty in paths but cannot create information that was not observed.