I have a question concerning the calculation of mean values in histograms when using negative event weights.

When I fill a TH1D using weights that can be negative (e.g. in MC@NLO data),
I find that the mean value of the histogram is calculated strangely.

I have identified the corresponding line in the source code of TH1D, which is copied below and marked at the respective lines with comments:

Int_t myTH1D::Fill(Double_t x, Double_t w)
{
...
Double_t z= (w > 0 ? w : -w);
// why is the weight here always put to be positive? Shouldn't it simply be z=w here?
fTsumw += z;
fTsumw2 += z*z;
// same point as above, should'nt it be something like
// if(w<0.)
// {
// fTsumw2+= (-1.)*(z*z);
// }
// else
// {
// fTsumw2+= z*z;
// };
fTsumwx += z*x;
fTsumwx2 += z*x*x;
return bin;
};

The problem becomes obvious when I try to scale the histogram by
histo->Scale(scalefactor);

The mean of the histogram changes here after scaling when using negative event weights, which should not be.

Maybe I didn’t understand quite well what negative weights really mean, but
according to the MCatNLO documentation, they should be treated as usual weights.

Best Regards,
Joerg Walbersloh EventWeights.C (397 Bytes)

I think you have touched a delicate point.
Negative weights are not really defined in a statistical sense and they are ambiguous. What is the mean of having for example two events in a bin with a weight of + 1 and then one event with a weight -1 ?
Is it equivalent to fill only one time with a weight of 1 or the error is the sum of the weight square, sqrt(3) ?
Also, calculating a weighted mean using negative weights does not make any sense at all.

Do you know what do they mean in the MC@NLO with the negative weights ?

a solution might be to create pairs of histograms: one for events with positive weights and a second one for those with negative weights (where you fill the events with |weight|). Then you do your histogram manipulations separately, and only combine them at the end. This should give you well-defined behavior for all intermediate operations. I have no idea whether this helps you in any way for the weighted mean, though - probably not. But maybe you can delay the combination of the pos and neg samples here, too. At least scaling them will be well defined.

I have instead created myself a solution by writing a class ‘myTH1’ which inherits from the original TH1. I have modified this new class to my purposes; I could identify all problems in the original source code that were related to the negative weights and redefined these code pieces to work with ±1 weights now.

I can use my own histogram via LoadClass now, but of course, this is only a temporary solution (and not the best, since I do not know what will happen if I use other methods with my modified statistics).
At least, I can do what I need right now.

Nevertheless, the main point on understanding the correct statistical meaning of negative weights still remains an open issue and shoul be discussed further.