Binary cross-entropy loss

The binary cross-entropy loss is given by:

$L=-{\frac {1}{m}}\sum _{i=1}^{m}\left[y_{i}\cdot \log {({\hat {y_{i}}})}+(1-y_{i})\cdot \log {(1-{\hat {y_{i}}})}\right]$

for $m$ training examples (indexed by $i$) where $y_{i}$ is the class label (0 or 1) and ${\hat {y_{i}}}$ is the prediction for that example (i.e. the predicted probability that it is a positive example). Thus $1-{\hat {y_{i}}}$ is the probability that it is a negative example.

True Positive

$\left[1\cdot 0+(0)\cdot -\infty \right]\approx 0$

False Negative

$\left[1\cdot -\infty +(0)\cdot 0\right]\approx -\infty$

False Positive

$\left[0\cdot 0+(1)\cdot -\infty \right]\approx -\infty$

True Negative

$\left[0\cdot -\infty +(1)\cdot 0\right]\approx 0$