[Rivet] Rivet "collaboration meeting"?

[Simon Plätzer]

Hi there,

David asked me to quote my understanding of this issue:

On Tue, 2013-05-21 at 21:23 +0200, Andy Buckley wrote:
> On 21/05/13 21:01, Leif Lönnblad wrote:
> > On 2013-05-21 16:50, Frank Siegert wrote:
> >> Why is this not using the normal variance a la
> >> s^2 = 1/(n-1) <(w-<w>)^2>
> >>     = 1/(n-1) (<w^2> - <w>^2)
> >>     = 1/(n-1) (sum(w^2)/n - sum(w)^2/n^2)
> > 
> > If we use this, the error will be zero for uniform weights...
> 
> Yes, this would be a measure of the width of the distribution of weights.

No. It's the variance of the weight mean; if you leave out a factor 1/n
it's the variance of the weights. sum(w^2)/n^2 clearly is the large
sample limit of Frank's formula.

I think what Frank has in mind (and what my recollection is, as well)
applies to the case of bins for a cross section differential in some
variable and one has to be careful what n is.

What I think of is a bin in a differential distribution, i.e.

(1)
d sigma / d O = \int_phi delta(O-Odef(phi)) dsigma(phi)

where Odef(phi) is the definition of the observable O in terms of a
phase space point phi and we integrate over phase space.

Then the MC incarnation of the integral above for a finite-size bin
(i.e. (1/Delta O) int_{O_-}^{O_+} d sigma / d O is

(2)
Delta sigma / Delta O = (sigma_tot / Delta O) (1/N) sum_{i=1}^N w(phi_i)
theta(O_+ - Odef(phi_i)) theta(O(phi_i) - O_-) = (sigma_tot / Delta O)
<w theta(event in bin)>

where I have inserted 1 = sigma_tot / sigma_tot into the right hand side
of (1) to introduce the total cross section as quoted by the event
generator, which will generate events with weights taken relative to
that cross section (the event generator samples events from
dsigma/sigma_tot and may do so by using weighted events or doing a
hit-and-miss to provide unit weights in the end).

Here, N is the _total_ number of events generated; to this definition,
the proper uncertainty is given by Frank's formula upon  letting n -> N
and multiplying by the appropriate normalization (and, there is no issue
for unit weights owing to the difference between the number of events in
the bin and the total number of events generated).

Best,
Simon