Changes between Version 1 and Version 2 of MatrixElement


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Timestamp:
04/06/12 16:33:02 (8 years ago)
Author:
trac
Comment:

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  • MatrixElement

    v1 v2  
    77The likelihood for N events is defined as %\[ L(\alpha)=e^{-N \int \bar{P}(x,\alpha)dx} \prod_{i=1}^{N} \bar{P}(x_i;\alpha)\]%
    88
    9 The 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 %$\alpha$%, %\[-ln (L)=-\sum_{i=1}^{N} ln(\bar{P}(x_i;\alpha)) + N \int \bar{P}(x,\alpha)dx\]%
     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 $\alpha$, %\[-ln (L)=-\sum_{i=1}^{N} ln(\bar{P}(x_i;\alpha)) + N \int \bar{P}(x,\alpha)dx\]%
    1010
    11 In 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 %$\alpha$%.
     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 %$P(x,\alpha)$%: %\[\bar{P}(x,\alpha){{{
     12Acc(x) P(x,\alpha)\]% where %$ Acc(x)$% is the detector acceptance, which depends only on %$ x $%. So the quantity that we have to minimize is %\[-ln (\tilde{L})
     13}}}-\sum_{i=1}^{N} ln(P(x_i;\alpha)) + N \int Acc(x) P(x,\alpha)dx\]% where the term $-\sum_{i=1}^N ln(Acc(x_i))$ has been omitted since it does not depend on $\alpha$.
    1214
    1315== Definition of the Weight ==
    1416
    1517The 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
     18   1. $ x $ is the set of information describing the events in the detector (momenta,tag,...)
     19   1. $ \alpha $ describe a theoretical hyppothesis
     20   1. $\sigma_{ \alpha}$ is the cross section of this theoretical hyppothesis
     21   1. $M_{ \alpha}$ is the aplitude linked to this theoretical framework
     22   1. $f_i(w_i)$ is the parton distribution function associate to the initial parton
     23   1. $W(x, y)$ is the TransferFunction
    2224
    2325== Computation of those elements ==
     
    2628
    2729   1. TransferFunction
    28    1. [:MadWeight:Computation of the Weight]
     30   1. [wiki:MadWeight Computation of the Weight]
    2931   1. AcceptanceTerm
    3032
     
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