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A Help with the statistics of Upper Limits?

  1. Apr 29, 2017 #1

    ChrisVer

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    Gold Member

    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:
    [itex]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)[/itex]
    Where [itex]N_{obs}[/itex] is the observed events, [itex]b/s[/itex] are the background/signal expected events, [itex]\theta_i[/itex] are the different nuisance parameters and [itex]\mu[/itex] 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 [itex]U(\mu)[/itex]- let's consider it Uniform. [itex]P(x|n)[/itex] is the poisson probability to get x observed given the expectation of n, and [itex]Gaus[/itex] 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 [itex]N_{obs}=N_{exp}=b[/itex].

    2. Once they do it, they can start varying the background+signal uncertainties, [itex]\theta_{stat}[/itex] (+[itex]\theta_{sys}[/itex]) [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 [itex]\mu[/itex] so that the [itex]b'+\mu s' = N_{obs}[/itex]

    4. Doing that several times, you get a distribution for [itex]\mu[/itex] (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 [itex]\theta_{sys}[/itex] on top of the statistical), moves the [itex]\mu_{.95}[/itex] 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. jcsd
  3. Apr 29, 2017 #2

    mfb

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    2017 Award

    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.
     
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