Changes between Version 2 and Version 3 of MatrixElement


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Timestamp:
Apr 12, 2012, 11:55:46 AM (13 years ago)
Author:
Martin
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  • MatrixElement

    v2 v3  
    55The Matrix Element Method consist in minimizing a likelihood.
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    7 The likelihood for N events is defined as %\[ L(\alpha)=e^{-N \int \bar{P}(x,\alpha)dx} \prod_{i=1}^{N} \bar{P}(x_i;\alpha)\]%
     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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    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 $\alpha$, $-ln (L)=-\sum_{i=1}^{N} ln(\bar{P}(x_i;\alpha)) + N \int \bar{P}(x,\alpha)dx$
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    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 %$P(x,\alpha)$%: %\[\bar{P}(x,\alpha){{{
    12 Acc(x) P(x,\alpha)\]% where %$ Acc(x)$% is the detector acceptance, which depends only on %$ x $%. So the quantity that we have to minimize is %\[-ln (\tilde{L})
    13 }}}-\sum_{i=1}^{N} ln(P(x_i;\alpha)) + N \int Acc(x) P(x,\alpha)dx\]% where the term $-\sum_{i=1}^N ln(Acc(x_i))$ has been omitted since it does not depend on $\alpha$.
     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 $\alpha$.
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    1515== Definition of the Weight ==
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    17 The 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
     17The 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
    1818   1. $ x $ is the set of information describing the events in the detector (momenta,tag,...)
    1919   1. $ \alpha $ describe a theoretical hyppothesis
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