Changes between Version 2 and Version 3 of MatrixElement
 Timestamp:
 04/12/12 11:55:46 (8 years ago)
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MatrixElement
v2 v3 5 5 The Matrix Element Method consist in minimizing a likelihood. 6 6 7 The likelihood for N events is defined as %\[ L(\alpha)=e^{N \int \bar{P}(x,\alpha)dx} \prod_{i=1}^{N} \bar{P}(x_i;\alpha)\]%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)$ 8 8 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 $\alpha$, %\[ln (L)=\sum_{i=1}^{N} ln(\bar{P}(x_i;\alpha)) + N \int \bar{P}(x,\alpha)dx\]%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 $\alpha$, $ln (L)=\sum_{i=1}^{N} ln(\bar{P}(x_i;\alpha)) + N \int \bar{P}(x,\alpha)dx$ 10 10 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 $\alpha$.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 $\alpha$. 14 14 15 15 == Definition of the Weight == 16 16 17 The Matrix Element Method associates a weight to each experimental event %\[ P( x  \alpha)=\frac{1}{\sigma_{ \alpha}} \int d \phi( y) M_{ \alpha}^2 ( y) dw_1 dw_2 f_1(w_1) f_2(w_2) W(x, y) \]%where17 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 18 18 1. $ x $ is the set of information describing the events in the detector (momenta,tag,...) 19 19 1. $ \alpha $ describe a theoretical hyppothesis … … 50 50 51 51 52