# Re: strange behavior of likelyhood fitting errors (fwd)

From: Lorenzo Moneta <lorenzo.moneta_at_cern.ch>
Date: Tue, 3 May 2005 11:32:38 +0200

Hi Sebastien,

if I understood well, you are performing a binned likelihood fit to an histogram.
This method assumes that you have counting events in your histograms distributed according to a Poisson distribution ( see for example the PDG review on statistics http://pdg.lbl.gov/2004/reviews/statrpp.pdf , 32.12) Applying a normalization (multiplying functions and data by a scale factor) has an effect on the result because the poisson probability (and therefore the likelihood) is not invariant if you multiply number of observed events and expectation values by a scale factor.

So, in conclusion, I would recommend either to use the correct countings on the bins contents, or even better use the unbinned likelihood method, where you fit directly your data sets, without going through the histogram.

In this last case you need to normalize your fitting function.

Best Regards

Lorenzo

If you applies normalization to the histograms, you have to be careful, to the statistical meaning of the bin contents, which can be different when you have variable bin histograms. Therefore it is not surprising that you get different results depending on the normalization.
The correct procedure is to normalize the function according to th

On May 3, 2005, at 8:17 AM, Rene Brun wrote:

> Lorenzo,
>
> Could you process this case?
>
> Rene
>
> ---------- Forwarded message ----------
> Date: Tue, 03 May 2005 01:45:43 +0200
> Subject: [ROOT] strange behavior of likelyhood fitting errors
>
> Hi rooters,
>
> I am puzzling with the errors returned by a likelyhood fitting made in
> two different ways.
>
> - First way, I plotted dimuon invariant mass data on a TH1 histogram
> with variable bining (first bins have width of 0.3 GeV then some with
> width equal to 0.6 and then a few with 0.9 GeV). I decided to
> normalize
> each bin by the size of the smallest bin (here is 0.3 GeV) so a bin
> represent the number of events per mass unit per 0.3 GeV. Then I apply
> my fit function (which is a sum of simple fits) on this histo by:
>
> hSigFitbgd->Fit(myFitFunction,"ILLEM","",xlow,xhigh);
>
> I use "LL" option to use log likelyhood minimisation for bin entries
> which are not integers (as I read in the documentation).
> I have three parameters in my fit function. I found for these three
> fit
> the follwing :
> First parameter : 489 ± 25
> Second parameter : 154 ± 45
> Third parameter : 270 ± 50
>
> Comparing these numbers to a Chi square method fit results looks
> reasonable except that I get smaller error especially for the second
> parameter :
> First parameter : 489 ± 30
> Second parameter : 154 ± 89
> Third parameter : 270 ± 57
>
> Since the statistics is low I assume the error are smaller because log
> likelyhood is more suited for low statistics sample.
>
> But now (second way of fitting), I just remove the normalisation by
> the
> smallest bin so a bin entry just is the number of events per mass unit
> (delta N/delta M). Now I just apply the exactly same fit function
> and I
> get very small error like :
> First parameter : 489 ± 11
> Second parameter : 155 ± 22
> Third parameter : 268 ± 26
>
> The two ways are exactly the same except for the building of the
> histogram to fit. It seems that the removal of the normalisation
> factor
> (actually the 0.3 normalisation factor) cause a impressive decrease of
> the error. I was wondering how the likelyhood method handle the error
> propagation... Does anyone have an idea ?