Version 1 (modified by trac, 7 years ago) (diff)

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## Matrix Element Method

The Matrix Element Method consist in minimizing a likelihood.

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 %% is obtained through a maximisation of the likelihood. It is common practice to minimize %% with respect to %%, %$-ln (L)=-\sum_{i=1}{N} ln(\bar{P}(x_i;\alpha)) + N \int \bar{P}(x,\alpha)dx$%

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 %% can be related to the produced probability density %%: %$\bar{P}(x,\alpha)Acc(x) P(x,\alpha)$% 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 %% has been omitted since it does not depend on %%.

## 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. %% is the set of information describing the events in the detector (momenta,tag,...)
2. %% describe a theoretical hyppothesis
3. %% is the cross section of this theoretical hyppothesis
4. %% is the aplitude linked to this theoretical framework
5. %% is the parton distribution function associate to the initial parton
6. %% is the TransferFunction

## Computation of those elements

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/

1. TransferFunction