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

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 ?

$$s$$ 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?

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

• Your explanation is interesting, but I don't see direct connection to my example. In an accelerator, at each collision, we have a process only (not several only). The process could be background or signal. But there could not be several processes within a single collision. So, there is no probability of having a given number process in a given event. Also, there is no time that enter in the game : we have discrete collisions, separate by a given time. At each collision, we see either a signal process, or a background process. If we integrate on N events, we will have s+b=n. But n is not rare ? Commented Apr 9, 2021 at 20:56
• @MathieuKrisztian in your question you suggested that there may be $n$ events... but now you say that there may be only $1$ event. For one event you need only the probability $p$ that event happens, and probability $q=1-p$ that it doesn't. If however you repeat experiment many times, you are back to the binomial distribution, and if $p\ll 1$, i.e., the event does not happen in every experiment, then the events are rare. Commented Apr 10, 2021 at 6:56
• the $n$ events correspond to the integration of all collisions : in one collision, there is one event, not more. All people that gave a comment made the hypothesis that in a collision, there are plenty of processes who happen and are recorded : this is completely wrong. There is no probability here. Is there somebody that knows both particle physics and statistics that could give a comment. It looks like people who reply know statistics but not experimental particle physics, so they give just "generic" answer, who have no relationship with the condition of the experiment. Commented Apr 10, 2021 at 7:34
• @MathieuKrisztian $s$ and $b$ are used here as probabilities. However, the average number of events in the Poisson law is $s+b$, so these terms might be used interchangeably. Commented Apr 10, 2021 at 9:44
• @MathieuKrisztian I added a remark regarding one possible confusion fo termonology here (since two different things are denoted by the same letters) Commented Apr 10, 2021 at 9:57

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.

• Sorry, I'm lost with your explanation. When we do collisions, all events are recorded, but we skip most of them because of selection choice. The poisson law $e^{-\lambda}\frac{\lambda^n}{n!}$ is assuming that $\lambda$=big number of events time low probability. The probability of background is not low : it is high. What you say is very important : are you meaning that there is no need of rarity ? See : en.wikipedia.org/wiki/Poisson_distribution#Law_of_rare_events What should be rare ? Commented Apr 6, 2021 at 20:13
• @MathieuKrisztian: No, this is not what I said. I add an example. Commented Apr 6, 2021 at 21:31
• I'm not sure at all with your explanation. At LHC there are $1e11$ protons in a bunch but in ost cases protons of two colliding bunches don't interact. The average number of interactions per bunch crossing is typically 40 for LHC Run 2. So in a single collision, there could be 40 processes. Your nulbers are much too big Commented Apr 7, 2021 at 6:34
• When a collision creates a process we have either signal or background but signal+background is not rare. Poisson usage require that number of collisions is large: this is the case, but also that probability of collision is low: this is not the case: each time a proton of a bunch interacts with a proton of the other bunch, it creates particles. So probability is not low. Commented Apr 7, 2021 at 6:37
• Also one process creates typically at most 100 particles, not 10000 Commented Apr 7, 2021 at 6:38