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| 3 | == Matrix Element Method == |

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| 5 | The Matrix Element Method consist in minimizing a likelihood. |

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| 7 | The likelihood for N events is defined as %\[ L(\alpha)=e^{-N \int \bar{P}(x,\alpha)dx} \prod_{i=1}^{N} \bar{P}(x_i;\alpha)\]% |

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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 %$\alpha$%, %\[-ln (L)=-\sum_{i=1}^{N} ln(\bar{P}(x_i;\alpha)) + N \int \bar{P}(x,\alpha)dx\]% |

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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 %$P(x,\alpha)$%: %\[\bar{P}(x,\alpha){{{Acc(x) P(x,\alpha)\]% 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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| 15 | 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 |

| 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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| 25 | The 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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