Uncertainty of histogram bins from MC-simulation

[minits]

TL;DR Summary

How to determine the uncertainty of histogram bins when the entries result from a MC-simulation

Hello everyone,

I calculated the matrix element of a parton level process and determined the total cross section via a MC-simulation. Then I wanted to look at some differential distributions like the differential cross section with respect to the energy of one of the particles in the final state. In order to do this I calculate the differential cross section for 10.000 phase-space points and then do a case analysis in which bin to add the respective result. Now I am stuck with assigning an uncertainty to each bin. How does this work? The differential cross sections do not carry any uncertainty since they are calculated analytically. One approach I often read is the use of the poissonian distibution but I do not think it is the proper way since I feel like I am not really doing a counting experiment. Thanks for any advice!

 

[Twigg]

Can you show us the histogram? If you want to protect your data, feel free to apply a blind to the axes. I'd just like a sense for how many points fall in each bin before I say something stupid.

I know next to nothing about the physics of what you're studying; however, I'm confused how you could be generating a histogram from MC results and not be doing a counting experiment? Can you explain why you think you're not doing counting? Where I'm going with this is that all histograms are counting. The Poissonian distribution tells you the estimated uncertainty on the number of MC results in each bin (in the appropriate limit of many MC results).

 

[minits]

Here is the histogram you asked for. I am just not firm with the application of the poisson distribution since everyone says it kind of always applies in counting experiments but I still don't know why that is.
A second thing that I am unsure about is if the error from the MC ist poissonian I should take sqrt(N) as the errors for each bin but I have 10.000 events and in every event another number flows into one of the bins. So if I have 100 events flowing into one bin and the value of these events sum up to two the error won't be 10. So probably I should interpret the specific values as a weight which comlicates things.

 

[Twigg]

Ok, seeing that I feel more confident saying Poisson is what you want.

I could give you my spiel about why the Poisson distribution is appropriate, but there are probably 1000's of folks online that have already given good explanations and put more time in than I would have. This video might be a good place to start.

The technical reason for why the poisson distribution is appropriate is that it is the limit of the binomial distribution as the number of trials goes to infinity and the mean number of successes is kept finite. Whenever you skim a few outcomes from a much larger pot of outcomes (raindrops on a roof, electrons over a threshold, photon shot noise, or points in a histogram bin), the Poisson distribution is what you need. If what you're asking is how you assign error bars to your histogram plot, take the square root of the number of points in each bin (that's the signature feature of the PD).

 

[mfb]

You are counting phase space points. If they are selected randomly then Poisson-like uncertainties should be right (your events have weights, so it's not a simple sqrt(N)). If they come from some grid then the uncertainty will be smaller. If computing power allows you can repeat the analysis with different grid patterns and compare the results to estimate the uncertainty.

Just from looking at your plot, bin-by-bin variations are probably very small.

 

[Twigg]

Whoops, I didn't pay attention to the y-axis of the histogram. @mfb is right. Since the events are weighted, the error is more complicated than a sqrt(N).

This resource might be useful, but it'll be up to you to see if it fits your data. The varied bin width analysis would be a good check

https://www.zeuthen.desy.de/~wischnew/amanda/discussion/wgterror/working.html

 

[minits]

Thanks for your replies! I actually took the info for the uncertainties from the webseite you linked. However I did not understand the following equations:
var(w_i * 1 event) = w_i^2 * var(1 event) = w_i^2
Do you know why the these three expressions are equal? In the first relation why why is it the square of the weight? And after the second equal sign why is the variance of 1 event 1 and not 0? They also talk about
"poissonian fluctuation of the number of events" but where is this used explicitly?

 

[Twigg]

I'm on my phone so forgive the lack of LaTeX.

The first equality is error propagation. Var[a*x] = a^2 Var[x]. The second is because for poissonian Var[N events] = N.