# Caveats when using event-by-event reweightings?

A fairly common technique in experimental particle physics is event-by-event reweighting. The idea is that you have a sample of background-model events, either from a Monte Carlo simulation, or from a data-driven method. However, your background model doesn't describe the data well enough, so you go into a sideband region (where the signal you are looking for is neglegible), and look at a distribution $x$ which is incidental to the variables you are actually interested in $y$. Then you "fix" the model to fit the data in this distribution. For example, if the background is too low at $x=x_0$, all events with that value of $x$ get a weight slightly higher than 1.

The result is that the distribution of $x$ is now almost perfectly modeled, and variables correlated to $x$ are hopefully better modeled than before, and you have hopefully not overfitted your background to mask a possible signal.

Reweighting often seems like cheating to new users, but you get used to it quickly and stop worrying - and if used correctly it can be indeed a valuable tool. Now, I'm wondering if there is an accepted set of (thumb) rules when and how you are allowed to do reweighting.

Points I'd like clarified:

• It seems $x$ and $y$ must not be too correlated. If $x = f(y)$ strictly, then it's cheating. If they are completely uncorrelated however, reweighting wrt. $x$ wouldn't change $y$.
• Signal must be neglegible in the region where the reweighting factors are determined, but what is neglegible? A ratio of S/B = 1e-6 (before cuts) seems OK, but what if all signal events are in that region?
• Sometimes the weights are determined bin-by-bin, $w,i = N_\mathrm{data,i}/N_\mathrm{BG,i}$. Sometimes you fit a function $w(x)$ to smooth out the weights. Any rules of thumb on how to determine the binning and/or the function?
• How to calculate the systematic uncertainties from the procedure? Is it enough to vary the $w_i$ or the fit parameters by their uncertainties?

Maybe there is also just a good reference you can point me to.

• I assume that you already understand how to apply the weights when processing the data and you just want to know how reliable the new weights are or are not, yes? – dmckee May 14 '13 at 19:07

• Say your interested in the dilepton invariant mass, so $y=M_{\ell\ell}$. You believe the angle between leptons is not well modeled, so you correct the $\varphi_{\ell\ell}$ MC distribution ($x$). You would do that in bins of $\varphi$, and you have one factor for each bin. One may determine the factor in a sideband, but applies it to all events. You seem to be thinking about normalizing to the Upsilon peak (~10 GeV), however in the case I mean the total normalization often doesn't change. (By the way, you're answer would better be a comment. We should move this discussion there.) – jdm May 12 '13 at 13:25
• (Seems you need 50 reputition to comment...) I was asking about the most general case, since this has many uses: Fixing MC, estimating BGs from data, estimating trigger effects without simulating the trigger, .... You give every event a different weight, and make a certain distribution $y$ match perfectly - but have no predictive power there anymode, since data=BG per construction. You hope that in a somewhat correlated distribution $x$ (maybe with a different selection) you now have a better modelling of the BG, but you'll be able to discern the signal, in the simplest case a bump. – jdm May 13 '13 at 8:19