Difference between revisions of "Binary cross-entropy loss"
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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] | 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] | ||
</math> | </math> | ||
| − | for <math>m</math> training examples (indexed by <math>i</math>) where <math>y_i</math> is the class label (0 or 1) and <math>\hat{y_i}</math> is the prediction for that example (i.e. the predicted probability that it is a positive example). Thus <math>1-\hat{y_i}</math> is the probability that it is a negative example. | + | for <math>m</math> training examples (indexed by <math>i</math>) where <math>y_i</math> is the class label (0 or 1) and <math>\hat{y_i}</math> is the prediction for that example (i.e. the predicted probability that it is a positive example). Thus <math>1-\hat{y_i}</math> is the probability that it is a negative example. |
| + | |||
| + | |||
| + | ====True Positive==== | ||
| + | :<math> | ||
| + | \left[ 1 \cdot 0 + (0) \cdot -\infty \right] \approx 0 | ||
| + | </math> | ||
| + | ====False Negative==== | ||
| + | :<math> | ||
| + | \left[ 1 \cdot -\infty + (0) \cdot 0 \right] \approx -\infty | ||
| + | </math> | ||
| + | ====False Positive==== | ||
| + | :<math> | ||
| + | \left[ 0 \cdot 0 + (1) \cdot -\infty \right] \approx -\infty | ||
| + | </math> | ||
| + | ====True Negative==== | ||
| + | :<math> | ||
| + | \left[ 0 \cdot -\infty + (1) \cdot 0 \right] \approx 0 | ||
| + | </math> | ||
Latest revision as of 15:54, 3 February 2023
The binary cross-entropy loss is given by:
![{\displaystyle 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]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2cc7893dd2d25f4f49ccea803f1a1a99b4940bc9)
for training examples (indexed by ) where is the class label (0 or 1) and is the prediction for that example (i.e. the predicted probability that it is a positive example). Thus is the probability that it is a negative example.
![{\displaystyle \left[1\cdot 0+(0)\cdot -\infty \right]\approx 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0325b9993888912cd9298656720fffc5adb6bc22)
![{\displaystyle \left[1\cdot -\infty +(0)\cdot 0\right]\approx -\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa4ed247125e66ac13d9eb2a7118afd2bc85afba)
![{\displaystyle \left[0\cdot 0+(1)\cdot -\infty \right]\approx -\infty }](https://wikimedia.org/api/rest_v1/media/math/render/svg/65fa3905551555abcf8c9635662474b1a3e18d7c)
![{\displaystyle \left[0\cdot -\infty +(1)\cdot 0\right]\approx 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8fdf9c6af38daea65606e3d086f46ef18c76eb3b)
