Review Poisson Point Process

1. History

An early application of the Poisson point process to photoelectron counting can be found in Mandel (1958)[1].

Mandel assumes that, conditioned on the instantaneous optical intensity $𝑃 (𝑡)$ , the probability of ejecting one photoelectron in a sufficiently small interval $[𝑡, 𝑡 + d 𝑡]$ is

𝑃 (d 𝑁 (𝑡) = 1 | 𝑃 (𝑡)) = 𝛼 𝑃 (𝑡) d 𝑡 + 𝑜 (d 𝑡)

This leads to a Poisson distribution for the number of photoelectrons in a finite time interval, conditioned on $𝑃 (𝑡)$ .

2. Poisson Point Process

Let $𝑁 (𝑡)$ denote the number of events observed in the interval $[0, 𝑡]$ .

A counting process $𝑁 (𝑡)$ is called an inhomogeneous Poisson process with intensity function $𝜆 (𝑡) \geq 0$ if $𝑁 (0) = 0$ , it has independent increments, and

\begin{matrix} 𝑁 (𝑡) - 𝑁 (𝑠) \sim Poisson (Λ (𝑠, 𝑡)) \\ Λ (𝑠, 𝑡) ≔ \int_{𝑠}^{𝑡} 𝜆 (𝑢) d 𝑢, 0 \leq 𝑠 < 𝑡 \end{matrix}

Here $𝜆 (𝑡)$ is the intensity or rate of the process.

𝐸 [d 𝑁 (𝑡)] = 𝜆 (𝑡) d 𝑡

Equivalently, for a sufficiently small interval $d 𝑡$ ,

\begin{aligned} 𝑃 (d 𝑁 (𝑡) = 1) & = 𝜆 (𝑡) d 𝑡 + 𝑜 (d 𝑡) \\ 𝑃 (d 𝑁 (𝑡) = 0) & = 1 - 𝜆 (𝑡) d 𝑡 + 𝑜 (d 𝑡) \\ 𝑃 (d 𝑁 (𝑡) \geq 2) & = 𝑜 (d 𝑡) \end{aligned}

The expected number of events in an interval $[𝑠, 𝑡]$ is

𝐸 [𝑁 (𝑡) - 𝑁 (𝑠)] = Λ (𝑠, 𝑡) = \int_{𝑠}^{𝑡} 𝜆 (𝑢) d 𝑢

For a homogeneous Poisson process $𝜆 (𝑡) = 𝜆$ , and therefore

𝑁 (𝑡) - 𝑁 (𝑠) \sim Poisson (𝜆 𝑡 - 𝜆 𝑠)

2.1. Event Times

Suppose $𝑛$ events are observed in $[0, 𝑇]$ at ordered times

0 < 𝑡_{1} < 𝑡_{2} < \dots < 𝑡_{𝑛} < 𝑇

For infinitesimal, mutually disjoint intervals around the observed events $[𝑡_{𝑖}, 𝑡_{𝑖} + d 𝑡_{𝑖}]$ , the probability of observing one event in every such interval is approximately

\prod_{𝑖 = 1}^{𝑛} 𝜆 (𝑡_{𝑖}) d 𝑡_{𝑖}

At the same time, the probability of observing no additional events in the remaining part of $[0, 𝑇]$ is

exp (- \int_{0}^{𝑇} 𝜆 (𝑡) d 𝑡)

Combining the two gives the joint probability density of the observed point configuration.

2.2. Likelihood

Observing events at times ${𝑡_{1}, 𝑡_{2}, \dots, 𝑡_{𝑛}}$ in the interval $[0, 𝑇]$ gives the point-process likelihood

𝐿 (𝜆) = exp (- \int_{0}^{𝑇} 𝜆 (𝑡) d 𝑡) \prod_{𝑖 = 1}^{𝑛} 𝜆 (𝑡_{𝑖})

The corresponding log-likelihood is

log 𝐿 (𝜆) = - \int_{0}^{𝑇} 𝜆 (𝑡) d 𝑡 + \sum_{𝑖 = 1}^{𝑛} log 𝜆 (𝑡_{𝑖})

The two terms have simple interpretations:

$- \int_{0}^{𝑇} 𝜆 (𝑡) d 𝑡$ penalizes the expected total number of events, while $\sum_{𝑖 = 1}^{𝑛} log 𝜆 (𝑡_{𝑖})$ rewards a model that assigns high intensity to the locations where events are actually observed.

Bibliography

[1] L. Mandel, Fluctuations of Photon Beams and their Correlations, Proceedings of the Physical Society 72, 1037 (1958), https://doi.org/10.1088/0370-1328/72/6/312.