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An artificial neural network maps a point in the space of input observables to
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some value of neural network output $x$. The neural network training error is
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given by equation~\ref{eq:NNerrorFunc}. A given point in the vector space
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spanned by the neural network input observables (denoted as ``feature space''
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in the following) contributes to the neural network training error $E$ by
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\begin{equation}
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E' = (1 - x)^2\cdot\rho^\tau + x^2\cdot\rho^{QCD}
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\end{equation}
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where $\rho^\tau (\rho^{QCD})$ denotes the training sample density of the $\tau$ signal
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and QCD--jet background at that point in feature space.
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The value $x$ assigned by the neural network to this region in feature space is
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determined by the requirement of minimal error:
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\begin{align}
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\frac{\partial E'}{\partial x} &= 0 \nonumber \\
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0 &= -2(1-x)\cdot\rho^\tau+2x\cdot\rho^{QCD} \nonumber \\
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x &= \frac{\rho^\tau} {\rho^\tau + \rho^{QCD}} \label{eq:probFracToX} \\
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\rho^\tau &= x(\rho^\tau + \rho^{QCD}) \nonumber \\
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\frac{\rho^{QCD}}{\rho^\tau} &= \frac{1}{x} - 1 \label{eq:rawTransformX}
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\end{align}
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N.B. that the ratio $\frac{\rho^{QCD}}{\rho^\tau}$ corresponds to the ratio of
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the normalized probability density functions of signal and background input
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observable distributions, i.e. $\int \rho^{\tau} d\vec x = 1$.
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In the case of multiple neural networks, one can derive a formula that maps the
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output $x_j$ of the neural network corresponding to decay mode $j$ according to
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the ``prior probabilities'' $p_j^\tau (p_j^{QCD})$ for true $\tau$ leptons (QCD
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jets) to pass the pre--selection criteria and be reconstructed with decay mode
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$j$.
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By substituting $\rho^s \rightarrow \rho^s p_j^s$ for $s \in \{\tau, QCD\}$ in
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equation~\ref{eq:probFracToX}, the output $x_j$ can be related to $p_j^\tau (p_j^{QCD})$
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by
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\begin{equation}
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x_j' = \frac{\rho^\tau \cdot p_j^\tau}
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{\rho^\tau \cdot p_j^\tau + \rho^{QCD} \cdot p_j^{QCD} }
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= \frac{p_j^\tau}
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{p_j^\tau + \frac{\rho^{QCD}}{\rho^\tau} \cdot p_j^{QCD} }
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\label{eq:probFracToXWithPriors}
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\end{equation}
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Substituting equation~\ref{eq:rawTransformX} into equation~\ref{eq:probFracToXWithPriors} yields
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the transformation of the output $x_j$ of the neural neural network corresponding to
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any selected decay mode $j$ to a single discriminator output $x_j'$ which for a given
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point on the optimal performance curve should be independent of $j$.
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\begin{equation}
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x_j' = \frac{p_j^\tau}
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{p_j^\tau + \left(\frac{1}{x_j}-1\right)\cdot p_j^{QCD} }
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\end{equation}
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