Changes between Initial Version and Version 1 of MatrixElement


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Timestamp:
03/20/12 16:15:51 (7 years ago)
Author:
trac
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  • MatrixElement

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     3== Matrix Element Method ==
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     5The Matrix Element Method consist in minimizing a likelihood.
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     7The likelihood for N events is defined as %\[ L(\alpha)=e^{-N \int \bar{P}(x,\alpha)dx} \prod_{i=1}^{N} \bar{P}(x_i;\alpha)\]%
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     9The best estimate of the parameter %$\alpha$% is obtained through a maximisation of the likelihood. It is common practice to minimize %$-ln(L(\alpha))$% with respect to %$\alpha$%, %\[-ln (L)=-\sum_{i=1}^{N} ln(\bar{P}(x_i;\alpha)) + N \int \bar{P}(x,\alpha)dx\]%
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     11In general, the probability that an event is accepted depends on the characteristics of the measured event, and not on the process that produced it. The measured probability density %$\bar{P}(x,\alpha)$% can be related to the produced probability density %$P(x,\alpha)$%: %\[\bar{P}(x,\alpha){{{Acc(x) P(x,\alpha)\]% where %$ Acc(x)$% is the detector acceptance, which depends only on %$ x $%. So the quantity that we have to minimize is %\[-ln (\tilde{L})}}}-\sum_{i=1}^{N} ln(P(x_i;\alpha)) + N \int Acc(x) P(x,\alpha)dx\]% where the term %$-\sum_{i=1}^N ln(Acc(x_i))$% has been omitted since it does not depend on %$\alpha$%.
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     13== Definition of the Weight ==
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     15The Matrix Element Method associates a weight to each experimental event %\[ P( x || \alpha)=\frac{1}{\sigma_{ \alpha}} \int d \phi( y) ||M_{ \alpha}||^2 ( y) dw_1 dw_2 f_1(w_1) f_2(w_2) W(x, y) \]% where
     16   1. %$ x $% is the set of information describing the events in the detector (momenta,tag,...)
     17   1. %$ \alpha $% describe a theoretical hyppothesis
     18   1. %$\sigma_{ \alpha}$% is the cross section of this theoretical hyppothesis
     19   1. %$M_{ \alpha}$% is the aplitude linked to this theoretical framework
     20   1. %$f_i(w_i)$% is the parton distribution function associate to the initial parton
     21   1. %$W(x, y)$% is the TransferFunction
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     23== Computation of those elements ==
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     25The MadWeight has created a series of tool to compute the transfer function, the weight, the cross-section, the likelihood,... Some of these tools have their own specific page/
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     27   1. TransferFunction
     28   1. [:MadWeight:Computation of the Weight]
     29   1. AcceptanceTerm
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     31-- Main.OlivierMattelaer - 22 May 2009
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