The number of selected events (M) in a sample should be binomially distributed with parameters: Number of tried events (N) and probability (p)
To make an EoT estimate from a biased sample with N events, we need to know how the probability in the biased sample differs from one in an inclusive sample
Many (most?) LDMX EoT estimates:
p(biased) = B p(inclusive)
With B being the biasing factor?
We can test it!
Alternative: p(biased) = W * p(inclusive),
With W ratio of average event weight of the two samples
For inclusive, the average event weight is just 1 :)
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Binomial basics
Valid for distribution corresponding to N binary yes/no questions
Sample with M selected events and N generated, the probability estimate is just M/N
So we want C = p(biased)/p(inclusive)
Ratio of two probability parameters isn’t usually well behaved
Except if the distribution is binomial :)
95% Confidence interval for this can be reliably calculated
CI[(ln(C)] = 1.96 sqrt{1/Ni - 1/Mi + 1/Nb - 1/Mb}
For a given C, we can extrapolate the 95% CI to a large enough sample size
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EcalPN
At 1e8, both hypotheses are OK
But the flat bias looks like it is in trouble
So we extrapolate from the ratio at 1e8 to 1e9
Note: loglog scale
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EcalPN
At 1e8, both hypotheses are OK
But the flat bias looks like it is in trouble
So we extrapolate from the ratio at 1e8 to 1e9
Note: loglog scale
At 1e9 events, flat bias is ruled out
Overestimates the biasing by 5% or so
We may need to generate larger samples
Weight estimate is still spot on (also holds for larger samples!)
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Kaon samples
We can always estimate the probability ratio correctly given two samples
But producing a large inclusive sample kills the point of biasing :)
For more complicated biasing procedures, we don’t have a good guess for what the biasing factor should be
Ideally it factorises with the PN bias but… it might not
The only handle we have is the event weight
Two sources:
PN bias and pair production down-bias:
Resampling photonuclear interactions
Weights are multiplied during an event
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Kaon samples
Weights are overestimating the bias :(
Issue with how we assign them?
For the LDCS production of the kaon sample, we can start a production with biasing factor being ~34