Initial-State Conditioning in Path Likelihoods

Why Separate the Starting State?

An event-history likelihood describes how plausible an observed sequence of transitions and periods without transitions is under a model. But a participant’s initial state can itself be random. For example, a cohort may include people who enter follow-up in different disease states, with probabilities governed by a separate model. The probability of entering in a particular state and the likelihood of the subsequent path are distinct contributions.

Separating these contributions clarifies what is being estimated. A likelihood conditional on the observed starting state uses information in the path after entry, but not information in how often each starting state occurred. A joint likelihood uses both. The distinction matters especially when the initial-state distribution and the transition intensities share parameters.

The time origin must be explicit: it might be diagnosis, enrollment, or another defined entry time. The initial state is the state recorded at that origin. A transition that occurred before entry is not part of the observed post-entry path; if the entry state is unknown, it cannot simply be treated as a known starting state.

Conditional Likelihood for a Transition Path

Consider a finite-state process with transition counting process \(N_{ij}(t)\), which counts observed transitions from state \(i\) to state \(j\), for \(i\ne j\). Let \(Y_i(t)=I\{X(t-)=i\}\) indicate that the subject is in state \(i\) just before time \(t\). Suppose the transition intensity is

where \(\mathcal F_{t-}\) is the observed history just before \(t\), and \(\alpha_{ij}(t;\theta)\) is the transition rate when the subject is in state \(i\). It may depend on time and on predictable covariates or history. This intensity specification means that, over a sufficiently short interval of length \(dt\), the conditional probability of an \(i\)-to-\(j\) transition is approximately \(Y_i(t)\alpha_{ij}(t;\theta)dt\).

For the usual single-jump path likelihood, assume the transition process has no simultaneous jumps, the intensities correctly describe the post-entry process given the observed history, and follow-up is observed through a known endpoint \(\tau\). Assume also that censoring or the observation endpoint is noninformative for the event-process likelihood, conditional on the information included in the model. In particular, an observation mechanism that depends on unobserved future transitions cannot be ignored. The path likelihood below is conditional on the initial state and on the observation scheme; it does not model how the subject came to have that initial state.

The product includes only factors at actual jumps: if no \(i\)-to-\(j\) transition occurs at a time, then \(\Delta N_{ij}(t)=0\) and that time contributes no jump factor. The integral accounts for the information in not seeing a transition. While a subject occupies state \(i\), the total outgoing rate is \(\sum_{j\ne i}\alpha_{ij}(t;\theta)\). In an absorbing state, all outgoing rates are zero.

Why the Likelihood Has This Form

The formula follows from the local meaning of an intensity. Over a short interval in which the subject is in state \(i\), the probability of no outgoing transition is approximately \(1-\sum_{j\ne i}\alpha_{ij}(t;\theta)dt\). Multiplying such probabilities across the intervals with no jump gives the exponential survival factor in the likelihood. At an observed transition from \(i\) to \(j\) at time \(t\), the transition-time density contributes \(\alpha_{ij}(t;\theta)\).

A more explicit justification divides follow-up into small intervals. On an interval of width \(dt\) with no jump, the log contribution is approximately \(\log\{1-\sum_{j\ne i}\alpha_{ij}(t;\theta)dt\}\), which is approximately \(-\sum_{j\ne i}\alpha_{ij}(t;\theta)dt\). Summing these terms and taking the limit gives the negative integrated outgoing rate. Each observed jump contributes the logarithm of its transition intensity. Thus the log-likelihood is

This expression is conditional on the specified initial state: the path after time zero is the object being modeled. If simultaneous jumps are possible, the single-jump product above is not automatically the correct contribution for a tied event; the joint jump structure must be specified. Likewise, a state recorded with error requires a model for that measurement process.

Factoring the Initial-State Distribution

The separation can be stated directly using the probability chain rule. Let \(Z\) denote covariates observed at entry, and suppose the model gives an initial-state probability \(\pi_{x_0}(\psi\mid Z)=P(X(0)=x_0\mid Z;\psi)\), where \(\psi\) indexes the initial-state model. If \(D\) denotes the observed post-entry path, then the joint likelihood factors as

For independent subjects, the corresponding factors multiply across subjects. If \(\psi\) and \(\theta\) are distinct parameters, the initial-state factor informs \(\psi\), while the conditional path factor informs \(\theta\). If the same parameter appears in both factors, omitting the initial-state term changes the likelihood for that parameter. Conditioning on \(X(0)=x_0\) is a deliberate change in the inferential target, not an algebraic step that is always harmless.

When the Initial State Is Not Observed

Conditioning on a specified starting state is not appropriate merely because the model includes states. If \(X(0)\) is unobserved, the likelihood for the observed path must account for uncertainty about it. For a discrete initial-state distribution, this means summing the state-specific path likelihoods weighted by their initial probabilities:

This is marginalization, not conditioning. The same principle applies if the initial state is known only imperfectly: a measurement or classification model may also be needed. The following calculation shows why choosing one state-specific likelihood without evidence can give a different answer.

Practical Checks and Common Errors

Before interpreting or maximizing a path likelihood, check that the state at time zero is defined and recorded, that the transition intensities apply after that time origin, and that the observed follow-up corresponds to the modeled observation scheme. If the state is uncertain, use a model that incorporates that uncertainty rather than assigning a state without justification. If follow-up begins only after a selection process, conditioning on the recorded state does not automatically account for selection or delayed entry; those mechanisms may require additional likelihood terms.

A frequent mistake is to write the initial-state probability into every likelihood even when the analysis is explicitly conditional on the observed initial states. The correction is to state the target first: the conditional path likelihood omits that factor, while the joint likelihood includes it when the initial state is modeled as random. The reverse mistake is also consequential: dropping \(\pi_{x_0}(\psi\mid Z)\) from a joint likelihood when it depends on the parameter being estimated discards information and changes the model.

A second common error is to call a conditional path likelihood a complete likelihood for the study. It is complete only for the post-entry event process under the stated conditioning and observation assumptions. It does not describe the frequency of initial states, the process before entry, or an informative censoring mechanism unless those components have been modeled.