Study of the BG yield bias from the optimization

I summarize here how I evaluated the size of the bias on the background yield coming from the fact that we use WS and sidebands both for the optimization and for the background subtraction.
Here are my assumptions:

we have about 1000 cuts which would be in principle equivalent except for the statistical fluctuations
the above cuts share 70% of the events (i.e. when moving from one cut to the other only 30% of the events are allowed to go in and out)
SWS,SBRS,SBWS and MC yields are independent poisson distributed variables
the means for SWS,SBRS,SBWS and MC (<-- SRS-SWS-a*(SBRS-SBWS)) are taken from data

The procedure is then the following:

Generate 1000 pseudo experiments, and 1000 different cuts for each of those
measure the significance (MC/sqrt(MC+...)) for those 1000 cuts
find among them the cut that gives the maximum significance
measure the difference between the estimated background (B=sws+a(sbrs-sbws)) and the true value of the background (the input to my random number generators)
plot this difference divided by sqrt(B): this tells me how many "sigmas" of bias we have on the background yield.

The distribution I obtain is approximately gaussian, with sigma ~1 and the following means:

*Size of biases on the background yield from the optimization process*
Channel	Bias (sigma)
Kpi	-1.6
K3pi	-1.7
Sat	-2.1
D+	-2.1

*From Ramon: size of biases on the measured (m1,m2) from the optimization process, compared to the statistical, systematic, and largest systematic from single source errors*
Variable	Delta	Stat	Syst	largest syst
m1	-0.02	0.16	0.08	0.06
m2	-0.10	0.69	0.20	0.13

In reality what we care about is by how much we have to scale up the estimated background distribution to correct for the optimization bias. This is what I will call in the following the "background scale". This picture shows how the background scale behaves as a function of the correlation assumed among the cuts (what was fixed above as 70%)
This is the Toy MC code
The next natural question is how reliable is this assumption of 70% correlation. The number was just a wild guess from my part with no justification at all.
What I did was then going back to the data samples we have and try to measure this number on data. I took the K1Pi sample, and measured the correlation of the first 1000 cuts ranked according to the MC+data "significance": what is reported in these plots is the fraction of events that each of the 1000 cuts has in common with the cut chosen for the analysis. The four different plots refer to SRS, SWS, SBRS and SBWS. As you can see the correlation varies in the range 60% - 70%. This reflects into a small change in the background scale factors:

*Size of the background scale factors when going from 60 to 70% correlation*
Channel	BCK scale at 60% correlation	BCK scale at 70% correlation
Kpi	1.31	1.27
K3pi	1.29	1.25
Sat	1.19	1.16
D+	1.1	1.1

These are the same correlations but measured this time on the dplus sample.

Alessandro Cerri, Last modified: Tue Sep 7 11:15:49 CDT 2004