# A Help with the statistics of Upper Limits?

1. Apr 29, 2017

### ChrisVer

This could as well go to the statistics, but I am looking at it from particle physics point of view...
Why adding systematic uncertainties worsen the expected upper limits to the signal strength?
I am trying to find where the flaw enters in the following logic:

0. The model most analyses use is the following likelihood:
$L( N_{obs} | b(\theta ) + \mu s(\theta ) ) = P(N_{obs} |b(\theta ) + \mu s(\theta ) ) U(\mu) \Pi_i Gaus(\theta_i | 0,1)$
Where $N_{obs}$ is the observed events, $b/s$ are the background/signal expected events, $\theta_i$ are the different nuisance parameters and $\mu$ is called the signal strength. In a Bayesian approach, one has to also to feed in a prior distribution for the signal strength parameter, which is the $U(\mu)$- let's consider it Uniform. $P(x|n)$ is the poisson probability to get x observed given the expectation of n, and $Gaus$ is a way to represent the variation of the nuisance parameters (given you have symmetric errors).

1. In order for one to get the expected limits, they would set $N_{obs}=N_{exp}=b$.

2. Once they do it, they can start varying the background+signal uncertainties, $\theta_{stat}$ (+$\theta_{sys}$) [these uncertainties don't affect the signal and background in the same way]

3. On the varied result, they would try to figure out what is the $\mu$ so that the $b'+\mu s' = N_{obs}$

4. Doing that several times, you get a distribution for $\mu$ (after you marginalize over the uncertainties) which is called the posterior pdf....

5. From μ-distribution get the 95-quantile point.

Now for some reason, adding nuisance parameters (such as $\theta_{sys}$ on top of the statistical), moves the $\mu_{.95}$ higher.
Is that because the uncertainties are not the same for bkg/signal?
Intuitively I can see how eg by subtracting the background from the observed result with higher uncertainties is going to give a more unclear picture of how much signal you allow in the game... but I don't see where this fits in the above logic.

2. Apr 29, 2017

### Staff: Mentor

Nuisance parameters make your µ distribution broader. Ignoring asymmetries in the Poisson distribution, they should not shift the expected µ (which should be zero if your method is sound), you just get another uncertainty that gets added in quadrature.