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 16:54, 3 February 2023
The binary cross-entropy loss is given by:
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\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] }
for Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle m} training examples (indexed by Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i} ) where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle y_i} is the class label (0 or 1) and Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \hat{y_i}} is the prediction for that example (i.e. the predicted probability that it is a positive example). Thus Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1-\hat{y_i}} is the probability that it is a negative example.
True Positive
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ 1 \cdot 0 + (0) \cdot -\infty \right] \approx 0 }
False Negative
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ 1 \cdot -\infty + (0) \cdot 0 \right] \approx -\infty }
False Positive
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ 0 \cdot 0 + (1) \cdot -\infty \right] \approx -\infty }
True Negative
- Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \left[ 0 \cdot -\infty + (1) \cdot 0 \right] \approx 0 }
