I am reading a paper from the ATLAS experiment on the identification of tau jets from background jets and came across this figure:

enter image description here

I am struggling to find what the formula is for the inverse background efficiency. Can someone explain to me how this is calculated?

Here is the link to the whole paper: http://cdsweb.cern.ch/record/2064383/files/ATL-PHYS-PUB-2015-045.pdf



2 Answers 2


A definition is given at the beginning of secion 5.3.

The performance of the tau identification algorithm in terms of the inverse background efficiency, 1/εbkg versus the signal identification efficiency, εsig is shown in Fig. 9. The identification efficiency is defined as the fraction of 1-prong (3-prong) hadronic tau decays that are reconstructed as 1-track (3-track) τhad−vis candidates, which also pass the BDT selection criteria. The efficiency is the product of the reconstruction efficiency defined in section 4.3 and of the identification efficiency.

I'm not an expert in this field. But it reads to me like the background efficiency is akin to a false positive probability, and the signal efficiency is akin to a true positive probability. So Fig. 9 is a kind of ROC curve, but with the inverse false positive probability instead of the false positive probability.


"Inverse [background] efficiency" (also known as "[background] rejection") is calculated as the reciprocal of [background] efficiency.

In more detail (using $\tau$ identification as example), let's say you have a collection of objects, a portion of which are true taus and the remaining portion of which are not taus. Apply your $\tau$ identification procedure; then the "signal efficiency" would be calculated as

\begin{equation} \varepsilon_\tau = \frac{\text{number of true $\tau$ identified as a $\tau$}}{\text{total number of true $\tau$}} \end{equation}

However, there will also be some fraction of the not-taus in the collection of objects that could look like a tau, and thus be identified as a tau. So similarly there will be a "background efficiency"

\begin{equation} \varepsilon_b = \frac{\text{number of not-$\tau$ identified as a $\tau$}}{\text{total number of not-$\tau$}} \end{equation}

and from this, the rejection value is calculated as $1/\varepsilon_b$.


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