The likelihood confusion

Roofit and RooStats
1

Hi,

I have some confusions to understand the likelihood. Hoping someone can help me.

Suppose I have some data ( and it’s Gaussian distribution like mh ), I want to estimate the error of measure this mh. I find that in the RooFit manual, the example is something like 1) fitting the signal&background with Gaussian&Exp distribution, 2) Once the sigma and mu got, the -2Ln L got.
My confusion is that, my intuition is that the more data I have, the more accuracy I get. But in this way, if I have more data and the sigma&mu doesn’t change, I don’t get better accuracy, that’s why?

Another confusion comes from another way to calculate the Likelihood. L=f(x1,theta1)f(x2,theta2)… which x1, x2, represents the bin 1, bin 2 data ( simulated ) and theta1, theta2 represent the bin 1, bin 2 data ( observed ). This is assuming each bin is a Poisson distribution.
My confusion is that: suppose at first we have 20 bins and I get a 1-sigma limit when -2Ln L + 1 with some data x1, x2, x3,… , What happens if I consider one more bin, the L = L (with 20 bins) * f(the 21th bin), so in this case, the original 1-sigma limit is not the 1-sigma limit because now the the difference is not 1 but 1* f(the 21th bin).

In the end, if I have some mass distribution data, what is the best way to estimate the error of detect the mass ? ( the best mass, and the mass range in 1-sigma, 3-sigma… )

2

Hi,

I think the best is to look at some statistics book for HEP or follow some tutorial on statistics, like the CERN academic trainings that are recorded.

But in brief, answering your questions:

  • more data you have will make the log-likelihood narrower and you will get smaller uncertainties, but the central values of the parameter will not change significatly

  • The rule -2Ln L + 1 applies to likelihood ratio (log-likelihood differences) and not absolute likelihood values.
    The likelihood values vary a lot experiment by experiment. See for example this notebook:
    https://github.com/lmoneta/stat-course-ipmu/blob/master/Session2/LikelihoodAsGoF.ipynb

Lorenzo

3

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