Adding bin-by-bin significances

Newbie
1

Dear experts,

This is a follow-up on the following thread which is now closed: Significance bin by bin so I am creating a new thread.

I have a similar construct with a background and a signal histogram, and I want to manually calculate an estimate of the Asimov significance. I am calculating the significance in the i-th bin using:
Si = si/sqrt(bi+uncbi^2),
where si, bi and uncbi, are the signal, background and uncertainty in the background yield, respectively.

Now I wish to quantify the significance from all the bins, with a goal to maximize this sum. From @moneta 's answer in the above thread, I see that it is not simply the sum in quadrature of Si’s, but instead should be distributed like a Chi^2 distribution. Given this, does it make sense, for example, to maximize the reduced Chi^2? I would then evaluate:
Sum(Si^2)/N, with N as the total number of bins, and then maximize this for different binning choices. Is this valid? Does this still work if I choose varying bin widths?

I shall appreciate any help in this regard!

2

Hi,
Maybe it was not clear in that post. What you need to do when combining many bins is to write the full likelihood for all the bins and then use it to compute the global significance.
In the example there I was assume you have a normal distribution for each bin and then the full log-likelihood is the sum of the log of the gaussian in each bin. This applies when the significance for each bin can be expressed as S(i) = s(i)/sqrt(b(i)).
In your case I think you have a Poisson distribution in each bin with an expected value (s+b) multiplied with a normal term expressing the uncertainty sigma(b) in each bin. You will have to make then a model adding the log of each bin likelihood.
This will be general and valid if you have fixed or variable bin widths

Lorenzo