## 7.2 Conditional Intensities

From the standpoint of modeling point process data, it is most useful to think of the *conditional intensity function*. Given the history of a point process \(H_t\) up until but not including \(t\), the conditional intensity \(\lambda(t\mid H_t)\) of a point process is defined as

\[ \lambda(t\mid H_t) = \lim_{h\downarrow 0}\frac{\mathbb{E}[N(t, t + h)\mid H_t]}{h} \] if the limit exists. We can intepret the conditional intensity function as giving us the instantaneous rate at which points are being dropped on the time line at time \(t\), given the history up until that point.

For point processes that are well-behaved, the conditional intensity function uniquely determines the probability structure of the point process.

For a stationary Poisson process, the conditional intensity function is a constant, i.e. \(\lambda(t\mid H_t)=\lambda\). Following the definition of a stationary Poisson process, the conditional intensity does not depend on the history and does not depend on the time \(t\).

If \(\lambda(t\mid H_t)=\lambda(t)\), so that the conditional intensity does not depend on the history \(H_t\) (but may depend on \(t\) itself), the process described by \(\lambda(t)\) is a non-stationary (or inhomogeneous) Poisson process.

In the most general case, \(\lambda(t\mid H_t)\) depends on the history encoded by \(H_t\). In this case, the function \(\lambda(t\mid H_t)\) is itself random because it depends on the event times \(t_1, t_2,\dots, t_{N(t)}\), which are themselves random.