Why is it legitimate to use the Poisson law for likelihood computation in particle physics? (background events are not rare)

Question

In experimental particle physics at colliders, there are a high number of collisions of incoming particles, for example protons at LHC. Once protons collide in a given collision event, those protons that interact will create one process in the detector. So in one collision, there is one process only. When physicists computing the likelihood to observe, integrated on the huge number of collisions, $n$ events, while expecting (from a theoretical model) $s$ signal events and $b$ background events, one uses the Poisson law:

$$\text{Prob}(n|s+b)=e^{-(s+b)}\frac{(s+b)^n}{n!}.$$

So $n$ is the total events really observed in the detector after several collisions. $s$ is the total events of the signal process that theory model predicts after several collisions. $b$ is the total events of the background process that theory model predicts after several collisions.

The question is: why actually are we allowed to use the Poisson formula?

The Poisson formula is expected to be valid for rare events. But here, there is no probability that enters in the problem. So what is rare ?

See : https://en.wikipedia.org/wiki/Poisson_distribution#Law_of_rare_events

$s$ is rare by definition, but $b$ is not rare: at each collision, we get mostly background events, so $b$ is not rare.

So why could we use Poisson law ? And what is rare?

Answer 1

The basis for deriving the Poisson law is the assumption that the events occur with a constant rate, $\lambda$ (although one can generalize to a varying rate). We can then divide the time interval $t$ into $N$ intervals of size $\Delta t = t/N$, with the probability of an event occurring in each interval being $\lambda \Delta t$, and calculate the probability of $n$ events occurring, which is given by the binomial law $$ P_n = {N \choose n}(\lambda \Delta t)^n(1 - \lambda \Delta t)^{N-n} $$ One then can then take different limits:

• $n \ll N$ will result in the Poisson Law
• if $n/N$ is finite we will have a gamma distribution or even a Gaussian limit (extending to negative axis).

From this we see what rare really means here: the number of events is countable, whereas the number of intervals in our derivation can be made infinitely big. This is distinct from variables distributed according to Gaussian or other continuous distribution, which can be thought of as sums of many small events, but where we do not have the ability to differentiate between these events.

Note another implicit assumption in the derivation: we have only one event at a time.

I could recommend as further reading the following books:

• [The statistical analysis of recurrent events][1]
• [Survival and event history analysis: a process point of view][2]

Update
The time in the above formulas does not have to be literally the clock time. We could consider, e.g.,

• the probability of collision, $p$, taking place in an experiment, while performing $N$ identical experiments
• the probability of a process of interest occuring in a collision, given $N$ collisions

In either case we can still resort to the binomial distribution $$ P_n={N\choose n}p^n(1-p)^{N-n} $$ The average number of evens/collisions/processes is then $$ \langle n\rangle = pN $$ so again, if $\frac{\langle n\rangle}{N}=p\ll 1$ we can pass to the Poisson limit.

Remark
If we are given a Poisson law with rate $\lambda$, $$ P_n=\frac{\lambda^n}{n!}e^{-\lambda} $$ the average/expectation/mean number of events is $$ \langle n\rangle = \sum_{n=0}^{+\infty}nP_n = \lambda $$ In other words, the mean number of events is equal to the rate of the the process. When modeling one would thus sometimes take an actual average number of events and use it as the rate in the Poisson law. [1]: https://doi.org/10.1007/978-0-387-69810-6 [2]: https://doi.org/10.1007/978-0-387-68560-1

Answer 2

In the Poisson formula the average value $\lambda = s + b$ does not have to be small. E.g. if we have many particles passing through a detector and most of them do not contribute to the obtained signal (background or real), the conditions for a Poisson process are satisfied. This is exactly why we use this distribution to model the signal.

Here an example: Suppose we generate $n=10^{11}$ particles in a single collision. For each of these particles we "flip a coin": the probability that the particle generates an "interesting signal" and thus is stored after filtering is $p= 10^{-7}$, and the probability that it generates a "boring" signal is $q=1-p$. Thus, that after filtering we keep only $10^4$ "interesting" signals. Although this number is large compared to one, the probability for a single, randomly chosen particle to generate an "interesting" signal is small. Thus, the Poisson distribution will provide a pretty good approximation. Furthermore, any other distribution I can think of is either impossible to use or is worst.