Multiplicative Intensity Models and Likelihood Construction

From an Intensity Model to a Likelihood

A multiplicative intensity model separates two ingredients that play different roles in an event history. The exposure factor describes how much opportunity for an event is present at each time; the rate function specifies the event rate per unit of that exposure. Building the likelihood then requires accounting for both the rate at each observed event and the rate accumulated across the full observation period.

Let \(N(t)\) be a counting process observed on \([0,\tau]\), and let \(\mathcal{F}_{t-}\) denote the information available just before time \(t\). Suppose its conditional intensity under parameter \(\theta\) is \(\lambda(t\mid\mathcal{F}_{t-};\theta)=Y(t)r(t;\theta)\), where \(Y(t)\) is a nonnegative, predictable exposure process and \(r(t;\theta)\) is a nonnegative rate function. Predictability means that \(Y(t)\) is determined by information available just before \(t\), not by whether an event occurs at \(t\).

Definition: A multiplicative intensity model has conditional intensity \[ \lambda(t\mid\mathcal{F}_{t-};\theta)=Y(t)r(t;\theta), \] where \(Y(t)\) is a predictable exposure factor and \(r(t;\theta)\) is the rate per unit of exposure. The corresponding likelihood for an observed path has an event contribution from \(Y(t)r(t;\theta)\) at each jump of \(N\), and an integrated-rate contribution over the whole observation interval.

The usual path-likelihood expression applies when the process is a simple, non-explosive counting process with an absolutely continuous intensity, the observation interval and initial condition are specified, and the observed covariate and exposure histories are treated as given. There must be no simultaneous jumps for this single-process expression. If censoring is used to define the end of observation, its mechanism must be ignorable for the event-parameter likelihood, or else its likelihood must be modeled as well. The intensity must be positive at observed jumps; an observed event at a time when the model assigns zero intensity is incompatible with that parameter value.

For a fixed observed path, the likelihood, up to factors not depending on \(\theta\), is

L(\theta)\ \propto\ \exp\left\{ \int_0^\tau \log\!\big[Y(t)r(t;\theta)\big]\,dN(t) - \int_0^\tau Y(t)r(t;\theta)\,dt \right\}.

The first integral is a sum over observed event times: if the jumps occur at \(t_1,\ldots,t_K\), it contributes \(\prod_{j=1}^K Y(t_j)r(t_j;\theta)\). The second integral is the cumulative intensity, including times with no event. Equivalently,

L(\theta)\ \propto\ \left\{\prod_{j=1}^K Y(t_j)r(t_j;\theta)\right\} \exp\left\{-\int_0^\tau Y(t)r(t;\theta)\,dt\right\}.

This is a likelihood for the observed event history, not merely for the event count. If \(Y(t)\) is known and does not depend on \(\theta\), its event-time factors can be omitted when writing a likelihood proportional in \(\theta\). They should still be checked: if \(Y(t)\) depends on \(\theta\), those factors cannot be discarded.

Why Both Contributions Appear

The event term rewards parameter values that assign a high intensity where events actually occurred. The integrated term accounts for all the time during which an event could have occurred. Without that second term, increasing a rate would always increase the event contribution, so the expression would not correctly represent the probability density of the observed path.

A short interval argument gives the two pieces. Over a small interval of width \(dt\), conditional on the history at its start, an event has probability approximately \(\lambda(t)dt\), and no event has probability approximately \(1-\lambda(t)dt\). For a path with one event in each interval indexed by an observed event time and no event in the other intervals, multiplying these contributions gives factors \(\lambda(t_j)\) at jumps and factors \(1-\lambda(t)dt\) between jumps. Taking the limit as the intervals become arbitrarily small uses \(\log(1-u)\approx-u\):

\prod_{\text{no-event intervals}}\{1-\lambda(t)dt\} \ \longrightarrow\ \exp\left\{-\int_0^\tau\lambda(t)\,dt\right\}.

Combining this survival contribution with the event factors yields the path likelihood above. This argument explains why an event contributes its intensity, while an interval without an event contributes through the negative integrated intensity.

Worked Example: A Constant Rate With Recurrent Events

Suppose one process is observed from time 0 to time 12. Its exposure is \(Y(t)=1\) throughout, and events occur at times 2, 5, and 9. Assume the model \(r(t;\theta)=\theta\), with \(\theta>0\) measured in events per unit time. The process is simple and the recorded event times are distinct, so the single-process likelihood applies.

There are three event factors, each equal to \(\theta\). The integrated intensity is \(\int_0^{12}\theta\,dt=12\theta\). Therefore,

L(\theta)\ \propto\ \theta^3\exp(-12\theta).

At \(\theta=0.20\) events per unit time, the likelihood value with the parameter-independent path-density factor suppressed is

L(0.20)\ \propto\ (0.20)^3\exp\{-(12)(0.20)\} =0.008\exp(-2.4) \approx 0.0007257.

The exponent uses the full 12 units of observation, not just the times after the first event or the sum of event times. The no-event periods before, between, and after the observed events all contribute to the integrated term.

Exposure and Rate Can Vary With Time

The exposure process can change over time, and the rate can have a separate time pattern. The likelihood formula handles both by evaluating the product \(Y(t)r(t;\theta)\) at events and integrating that same product over the observation window. In practice, this often means splitting the integrated term into intervals on which the exposure or rate has a simple form.

Worked Example: Changing Exposure With a Constant Rate

Consider a recurrent-event process observed for 10 time units. Suppose its known, predictable exposure is \(Y(t)=2\) from time 0 to 4 and \(Y(t)=1\) from time 4 to 10. There are events at times 1, 3, and 7, and the rate per unit of exposure is constant, \(r(t;\theta)=\theta\). This exposure history is fixed by the observation plan, so it does not introduce additional unknown parameters.

At the three event times, the exposure factors are 2, 2, and 1. The integrated exposure is \(\int_0^{10}Y(t)\,dt=(2)(4)+(1)(6)=14\). Hence the likelihood, retaining the known event-exposure factors, is

L(\theta)\ \propto\ (2\theta)(2\theta)(1\theta)\exp(-14\theta) =4\theta^3\exp(-14\theta).

At \(\theta=0.20\) events per exposure-unit-time,

L(0.20)\ \propto\ 4(0.20)^3\exp\{-(14)(0.20)\} =0.032\exp(-2.8) \approx 0.001946.

Because the event-exposure factors are known and parameter-independent, the likelihood can also be written proportionally as \(\theta^3\exp(-14\theta)\). Omitting them does not change comparisons among values of \(\theta\), but the integrated exposure of 14 must remain.

Worked Example: A Piecewise-Constant Rate

Suppose a process is observed from time 0 to 8 with \(Y(t)=1\). Its rate is \(\alpha\) on \([0,4]\) and \(\beta\) on \((4,8]\), where both parameters are positive. The observed events are at times 1 and 6. These event times fall in different rate intervals, and the complete observation window is available.

The event at time 1 contributes \(\alpha\); the event at time 6 contributes \(\beta\). The integrated rate is \(4\alpha+4\beta\), since each rate applies over four time units. Thus,

L(\alpha,\beta)\ \propto\ \alpha\beta\exp(-4\alpha-4\beta).

For \(\alpha=0.08\) and \(\beta=0.04\) per unit time, the likelihood value is

L(0.08,0.04)\ \propto\ (0.08)(0.04)\exp\{-(4)(0.08)-(4)(0.04)\} =0.0032\exp(-0.48) \approx 0.001980.

The two event factors use the rate for the interval containing each event. The integrated term, in contrast, includes both rate intervals, including periods with no observed event.

Multiple Subjects and Censoring

For conditionally independent subjects with fixed observation windows, the joint likelihood is the product of their path likelihoods. Equivalently, their log-likelihood contributions add. A subject who is censored without an event contributes no event factor at the censoring time, but their observed at-risk exposure still contributes to the integrated intensity up to that time. This statement assumes the censoring mechanism is ignorable for the event model.

Worked Example: An Event and a Censored Subject

Two independent subjects have constant known exposure multipliers \(Y_A=2\) and \(Y_B=0.5\), respectively, and share the rate per exposure unit \(\theta\). Subject A has an event at time 3, which ends follow-up for that subject. Subject B is event-free and censored at time 4. Assume independent censoring for the event model and that both subjects enter observation at time 0.

Subject A contributes event factor \(2\theta\) and integrated intensity \((2)(3)\theta=6\theta\). Subject B contributes no event factor, but contributes integrated intensity \((0.5)(4)\theta=2\theta\). Multiplying the contributions gives

L(\theta)\ \propto\ (2\theta)\exp(-6\theta)\exp(-2\theta) =2\theta\exp(-8\theta).

At \(\theta=0.10\) per exposure-unit-time,

L(0.10)\ \propto\ 2(0.10)\exp\{-(8)(0.10)\} =0.2\exp(-0.8) \approx 0.08987.

The censored subject matters because the process remained under observation and event-free for four time units. Treating that subject as if follow-up ended at time zero would discard the \(\exp(-2\theta)\) contribution.

Assumptions, Diagnostics, and Alternatives

Before using the likelihood, check that the observation scheme matches its construction. The exposure must be predictable; the rate and exposure must define a valid nonnegative, integrable intensity; and the event times, covariates, and observation windows must be recorded on a time scale consistent with that intensity. For the simple-process formula, simultaneous events are not allowed. If events are observed when \(Y(t)=0\) or \(r(t;\theta)=0\), the likelihood is zero for that parameter value. If exposure or censoring depends on unobserved event-related information, treating it as fixed may not yield the intended likelihood.

Useful checks are direct rather than purely numerical: verify every event factor against the exposure and rate in force immediately before that event; independently sum the integrated intensity over all observed intervals; and check that subjects contribute only up to their actual end of follow-up. A very large event contribution paired with a small integrated term can flag omitted exposure time, while an integrated term that continues after censoring can flag an incorrectly extended observation window. These checks do not establish that the model is scientifically adequate, but they can reveal errors in its likelihood construction.

The path likelihood is a full likelihood based on event times and observed exposure. It differs from approaches that condition on selected parts of the data or use a discrete-time approximation. Such alternatives may be useful under their own assumptions, but they do not generally produce this same likelihood. In particular, replacing continuous-time event times by interval-level event indicators changes the observation model. The multiplicative form is also not a guarantee of a correct model: misspecified rates, unrecorded exposure changes, or informative censoring require additional modeling or a different inferential strategy.

The central practical rule is to keep the two contributions conceptually separate until the final expression is assembled: evaluate the intensity at every observed jump, then integrate it across every observed interval. That separation makes it easier to spot omitted events, lost exposure, and incorrect time limits.

Key takeaway: Under a multiplicative intensity \(\lambda(t)=Y(t)r(t;\theta)\), each observed event contributes its intensity at that time, and the entire observed period contributes the exponential of minus the integrated intensity. Known exposure factors may be dropped only when they are parameter-independent; their effect on integrated exposure must still be included.

Check Your Understanding

Answer each question, then compare with the solutions.

If \(Y(t)=3\) for six time units, the rate is \(\theta\), and there are two events, what are the integrated exposure and the likelihood up to parameter-independent factors?
A subject is event-free until an independent censoring time \(c\). Which likelihood contribution represents the observed event-free time?
For a piecewise rate \(r(t)=\alpha\) on \([0,2]\) and \(r(t)=\beta\) on \((2,5]\), with \(Y(t)=1\) and one event in each interval, write the likelihood.
Why can a known event-time exposure factor be omitted from a likelihood proportional in \(\theta\), while its contribution to the integrated intensity cannot be omitted?

Solutions. (1) The integrated exposure is \(3(6)=18\), so \(L(\theta)\propto\theta^2\exp(-18\theta)\). (2) The event-free period contributes \(\exp\{-\int_0^cY(t)r(t;\theta)\,dt\}\), with no event factor at \(c\). (3) The event factors are \(\alpha\) and \(\beta\), and the integrated rate is \(2\alpha+3\beta\), giving \(L(\alpha,\beta)\propto\alpha\beta\exp(-2\alpha-3\beta)\). (4) A known factor at an event is constant in \(\theta\), so it does not change likelihood comparisons across \(\theta\). Exposure also determines the integrated rate, which generally depends on the model parameter and represents observed event-free time, so it must be retained.