## 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.