Conditional and Marginal Intensities

Why the Information Set Changes an Intensity

An event rate is conditional on what is known just before the event could occur. If a model includes a latent risk type, its intensity describes event occurrence conditional on that type and the rest of the enlarged history. An analyst who does not observe the type instead needs the rate conditional on the observed history. These rates generally differ, even when the latent-type model is correctly specified.

The distinction matters because a rate averaged over a fixed population distribution is not usually the rate for people who remain event-free or have experienced a particular event history. Those histories change the distribution of latent types among the observed risk set. The observed-history intensity must account for that change.

The conditional expectation is taken at each time using the information available just before that time. It is not, in general, an average over the latent variable using its baseline distribution. Conditional on observed event history, the relevant distribution of the latent variable may have changed.

Why Conditional Expectation Gives the Observed Intensity

Suppose \(N(t)\) is adapted to the observed filtration and has enlarged-filtration intensity \(\lambda^{\mathcal G}(t)\). Assume the compensator is absolutely continuous in time, the intensity is locally integrable, and the conditional expectation has a predictable version. For any bounded, nonnegative, \((\mathcal F_t)\)-predictable process \(H(t)\), the compensator property gives

Because \(H(t)\) uses only the observed history just before \(t\), conditioning the integrand on that history yields

Thus the observed-history compensator has rate \(E\{\lambda^{\mathcal G}(t)\mid\mathcal F_{t-}\}\); uniqueness of the compensator identifies this as the observed-filtration intensity, up to the usual equality almost everywhere. This argument requires that the same counting process is adapted to the smaller filtration. If the event itself is not observed, its observed-filtration intensity is not obtained by this formula without first specifying an observed event process.

Survival and Event History Update the Average

A simple latent-class model shows why marginalization often changes the shape of an intensity model. Let \(Z\) be an unobserved type with \(P(Z=0)=P(Z=1)=0.5\). Suppose the first event time has an exponential distribution conditional on type, with rates \(0.5\) and \(1.5\) per month, respectively. The enlarged filtration includes \(Z\); the observed filtration includes the event history but not \(Z\). Assume everyone is initially at risk, the event is observed, and there is no other censoring or entry process in this calculation.

Among people who have had no event by time \(t\), Bayes’ rule gives a posterior probability of the higher-rate type proportional to its prior probability times its event-free probability:

Consequently, while still event-free, the observed-history intensity is the posterior mean of the type-specific rates. At \(t=0\), it is the baseline average \(0.5(0.5)+0.5(1.5)=1.0\) event per month. At later times, the higher-rate type is less common among those who remain event-free.

More generally, if \(S_z(t)\) is the event-free survival probability for type \(z\), the marginal survival function is \(S(t)=\sum_zP(Z=z)S_z(t)\). Its hazard is

The equality follows by differentiating \(S(t)\): since \(S'_z(t)=-\lambda_z(t)S_z(t)\), we have \(-S'(t)/S(t)=\sum_zP(Z=z)S_z(t)\lambda_z(t)/S(t)\), which is the stated posterior-weighted average. The weights depend on \(t\) whenever types have different survival distributions. Thus even constant type-specific hazards can produce a time-varying marginal hazard.

Cause-Specific Intensities Under Coarsened Information

For multiple event types, let \(N_k(t)\) count events of type \(k\), and let \(\lambda_k^{\mathcal G}(t)\) be its enlarged-history intensity. If the event types cannot jump simultaneously and each count is observed, the corresponding observed-history intensity is

Linearity of conditional expectation means the total intensity is also the sum of the observed cause-specific intensities. But the cause-specific rates can respond differently to changing beliefs about unobserved type. A constant cause-specific form conditional on type therefore need not remain constant after type is hidden.

What the Averaging Does—and Does Not—Preserve

Conditional expectation preserves sums and nonnegative weighted averages, but it does not generally preserve a model’s functional form. If an enlarged-history intensity is \(\lambda^{\mathcal G}(t)=Y(t)\lambda_0(t)\exp\{\beta Z\}\), where \(Y(t)\) is an observed at-risk indicator and \(Z\) is unobserved, then while at risk

The final factor is a history-dependent conditional expectation. It need not be constant, and it need not be expressible as an exponential of the same covariates with the same coefficient. Treating it as \(E\{\exp(\beta Z)\}\) ignores the information contained in the event-free and event history. Conversely, a model specified directly for the observed filtration need not imply a simple model conditional on latent variables.

This distinction also separates an observed-history intensity from a population-standardized summary. The former conditions on each person’s observed history just before time \(t\). A standardized rate or risk may instead average predictions over a chosen covariate distribution. The target, conditioning information, and averaging distribution should be stated before comparing estimates.

Assumptions, Diagnostics, and Common Pitfalls

The projection result is useful when the observed process is genuinely adapted to the observed filtration, the enlarged intensity is well defined, and the needed conditional expectation is integrable and predictably measurable. In applications, the observation scheme must also be described correctly: if event types, covariates, or censoring indicators are not observed as assumed, the filtration and resulting intensity change. If entry or censoring affects which histories are observed, those mechanisms must be handled in the model rather than silently treated as irrelevant.

A useful diagnostic question is whether the estimated rate is compatible with the information being conditioned on. If a fitted observed-history model assumes a constant rate, inspect whether rates or residual patterns vary with time since entry, event-free duration, or recorded event history. Such patterns may indicate model misspecification or unmodeled heterogeneity, but do not by themselves prove that a particular latent type exists. Standard residual checks can reveal some lack of fit; they cannot identify an unobserved mixing distribution from the observed data without additional assumptions.

• Wrong: “The marginal intensity is the baseline average of the type-specific intensities.” Correction: It is the conditional average given \(\mathcal F_{t-}\); the relevant type probabilities can change with observed history.
• Wrong: “A constant conditional rate implies a constant observed-history rate.” Correction: Different latent types can be selectively represented in the histories still at risk, making the observed rate time-dependent.
• Wrong: “The projected intensity retains the same regression coefficients.” Correction: Conditional expectation generally changes the functional form; the new intensity must be derived or modeled for the observed filtration.

A conditional model is appropriate when the scientific target is a rate given the more complete information and the relevant variables are measured or modeled. An observed-history model is appropriate when prediction or inference conditions only on the available history. A joint latent-variable model can represent both levels, but its conclusions rely on assumptions about the latent distribution and its relation to event occurrence. None of these approaches makes the other rate universally preferable; they answer different conditional questions.