Binomial Error Weighted Events
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sum {w_i} (i=1,N). The error on sum_w is then given as err(sum_w) = sqrt( sum {w_i^2} ). Derivation of the formula. Derivation of above formula is based weighted binomial distribution on error propagation and intrinsic poissonian statistics only. The variance var(sum_w) of sum_w (i.e. binomial error bars the "error on the weighted number of events" in that bin) is given by error propagation (err(sum_w))^2 == var(sum_w) = binomial standard deviation sum {var(w_i)} (i=1,N) , i.e. adding the squares of the errors on the weighted events. The variance var(w_i) of weight w_i is determined only by the statistical fluctuation of the number of events considered,
What Is Error Histogram
var(w_i) = var(w_i * 1 event) = w_i^2 * var(1 event) = w_i^2, with poissonian fluctuation of the number of events ("1 event"), and taking w_i to be a constant for event i. If this sounds difficult at first glance, just make the exercise and construct error propagation where you have 100 events split into two groups, with 90 events w_i==1.0 and 10 events with w_i==0.1 . We error histogram in neural network have sum_w = 90*1 + 10*0.1 = 91 events, the statistical fluctuation is coming from sqrt(90) and sqrt(10), giving var(w_i) = 1^2 * 90 + 0.1^2 * 10 = 90.1 , i.e. the error on sum_w is sqrt(90.1) = 9.49 . Your relative error is 9.49/91 = 0.105. Number of Equivalent Events. The number of equivalent events is defined as N_equ = ( sum_{w_i} )^2 / sum {w_i^2} . This number relates the sample of N weighted events to N_equ events with w==1 that would have the same relative statistical fluctuation. For the example above: The number of equivalent events there is N_equ = (sum_w)^2 / var(w_i) = 91.9 events. This means your statistics fluctuation is about as good (or bad) as for 92 events with event-weight==1. For the MC-files for the atm-nu's in the Nature paper: For 7000 events we get N_equ=2200, while the number of events in the data sample is 188. So, the "equivalent" statistics of MC is only about 12 times the data ! In certain regions of variable space, or for different distribution functions of the weights (which is the relevant quantity here !! ) you can be much better or worse ! Note also, that the oth
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