# Changes between Version 1 and Version 2 of MatrixElement

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

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 v1 The likelihood for N events is defined as %$L(\alpha)=e^{-N \int \bar{P}(x,\alpha)dx} \prod_{i=1}^{N} \bar{P}(x_i;\alpha)$% 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 %$\alpha$%, %$-ln (L)=-\sum_{i=1}^{N} ln(\bar{P}(x_i;\alpha)) + N \int \bar{P}(x,\alpha)dx$% 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 $\alpha$, %$-ln (L)=-\sum_{i=1}^{N} ln(\bar{P}(x_i;\alpha)) + N \int \bar{P}(x,\alpha)dx$% 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 %$\alpha$%. 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 $\alpha$. == Definition of the Weight == 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 1. %$x$% is the set of information describing the events in the detector (momenta,tag,...) 1. %$\alpha$% describe a theoretical hyppothesis 1. %$\sigma_{ \alpha}$% is the cross section of this theoretical hyppothesis 1. %$M_{ \alpha}$% is the aplitude linked to this theoretical framework 1. %$f_i(w_i)$% is the parton distribution function associate to the initial parton 1. %$W(x, y)$% is the TransferFunction 1. $x$ is the set of information describing the events in the detector (momenta,tag,...) 1. $\alpha$ describe a theoretical hyppothesis 1. $\sigma_{ \alpha}$ is the cross section of this theoretical hyppothesis 1. $M_{ \alpha}$ is the aplitude linked to this theoretical framework 1. $f_i(w_i)$ is the parton distribution function associate to the initial parton 1. $W(x, y)$ is the TransferFunction == Computation of those elements == 1. TransferFunction 1. [:MadWeight:Computation of the Weight] 1. [wiki:MadWeight Computation of the Weight] 1. AcceptanceTerm