  The conditional intensity (CI) of a counting process $Y_t$ is based on the
minimal knowledge $\mathcal{F}_t^Y$, i.e., on the observation of $Y_t$ alone.
Prominently, the mutual information rate of a signal and its Poisson channel
output is a difference functional between the CI and the intensity that has
full knowledge about the input. While the CI of Markov-modulated Poisson
processes evolves according to Snyder's filter, self-exciting processes, e.g.,
Hawkes processes, specify the CI via the history of $Y_t$. The emergence of the
CI as a self-contained stochastic process prompts us to bring its statistical
ensemble into focus. We investigate the asymptotic conditional intensity
distribution (ACID) and emphasize its rich information content. We assume the
case in which the CI is determined from a sufficient statistic that progresses
as a Markov process. We present a simulation-free method to compute the ACID
when the dimension of the sufficient statistic is low. The method is made
possible by introducing a backward recurrence time parametrization, which has
the advantage to align all probability inflow in a boundary condition for the
master equation. Case studies illustrate the usage of ACID for three primary
examples: 1) the Poisson channels with binary Markovian input (as an example of
a Markov-modulated Poisson process), 2) the standard Hawkes process with
exponential kernel (as an example of a self-exciting counting process) and 3)
the Gamma filter (as an example of an approximate filter to a Markov-modulated
Poisson process).
