# Changes between Version 2 and Version 3 of MatrixElement

Ignore:
Timestamp:
04/12/12 11:55:46 (8 years ago)
Comment:

--

### Legend:

Unmodified
 v2 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 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 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