Changes between Version 4 and Version 5 of MatrixElement

Timestamp:

Jun 27, 2015, 12:45:00 AM (9 years ago)

Author:

Olivier Mattelaer

Comment:

--

MatrixElement

@@ -5 +5 @@
 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)=\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 $\alpha$, $-ln (L)=-\sum_{i=1}^{N} ln(\bar{P}(x_i;\alpha)) + N \int \bar{P}(x,\alpha)dx$
 
-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 ==