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Binary log loss scores probabilistic predictions for labels 0 and 1 and strongly penalizes confident errors.
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Formula
nats per example with natural logarithms; dimensionless mean
| Symbol | Meaning | Unit |
|---|---|---|
| True binary label | 0 or 1 | |
| Predicted probability for label 1 | 0 to 1 | |
| Number of evaluated examples | examples |
L=-\frac1N\sum_{j=1}^{N}[y_j\ln(p_j)+(1-y_j)\ln(1-p_j)]For each example, y=1 leaves −ln(p), while y=0 leaves −ln(1−p). Confident wrong probabilities are therefore penalized heavily. Probabilities must be numerically clipped to avoid log(0); libraries implement such clipping. The loss evaluates probabilistic quality, not classification accuracy at one fixed threshold.
For each example, y=1 leaves −ln(p), while y=0 leaves −ln(1−p). Confident wrong probabilities are therefore penalized heavily. Probabilities must be numerically clipped to avoid log(0); libraries implement such clipping. The loss evaluates probabilistic quality, not classification accuracy at one fixed threshold.
Binary log loss scores probabilistic predictions for labels 0 and 1 and strongly penalizes confident errors.
Defining relation
The equation states the defining relation within its stated domain.
Use the example for Binary log loss to explain which inputs determine the requested output.
The example connects concrete inputs to its result.
Binary log loss scores probabilistic predictions for labels 0 and 1 and strongly penalizes confident errors.
The relation applies only under its stated assumptions.
Applying the relation to a different case without checking its stated assumptions.
Check the validity range before substituting values: For each example, y=1 leaves −ln(p), while y=0 leaves −ln(1−p). Confident wrong probabilities are therefore penalized heavily. Probabilities must be numerically clipped to avoid log(0); libraries implement such clipping. The.
Formula cluster
Binary log loss scores probabilistic predictions for labels 0 and 1 and strongly penalizes confident errors.
For y=1 and p=0.8, the individual loss is −ln(0.8)≈0.223 nats. At p=0.01 it rises to −ln(0.01)≈4.605 nats.
Train logistic regression and binary neural classifiers using their predicted probabilities.
Here is how to work through a typical Binary log loss (L = mean binary cross-entropy) task step by step:
Task
Use the example for Binary log loss to explain which inputs determine the requested output.
Solution path
The example connects concrete inputs to its result.
Binary log loss scores probabilistic predictions for labels 0 and 1 and strongly penalizes confident errors.
The relation applies only under its stated assumptions.
The relation L=-\frac1N\sum_{j=1}^{N}[yj\ln(pj)+(1-yj)\ln(1-pj)] summarizes Binary log loss scores probabilistic predictions for labels 0 and 1 and strongly penalizes confident errors. Read it as a model for defined quantities, not as an automatic claim about every real situation. In the example For y=1 and p=0.8, the individual loss is −ln(0.8)≈0.223 nats. At p=0.01 it rises to −ln(0.01)≈4.605 nats., symbols are connected to concrete values and the result is interpreted in context. Validity limit: For each example, y=1 leaves −ln(p), while y=0 leaves −ln(1−p). Confident wrong probabilities are therefore penalized heavily. Probabilities must be numerically clipped to avoid log(0); libraries implement such clipping. The Source: scikit-learn API Reference, log_loss.
First write down the requested quantity and all given values with their symbols. Substitute them into the defining relation and check unit consistency. The example For y=1 and p=0.8, the individual loss is −ln(0.8)≈0.223 nats. At p=0.01 it rises to −ln(0.01)≈4.605 nats. provides a worked check: formula and result must have compatible scale and meaning; round only at the end. Validity limit: For each example, y=1 leaves −ln(p), while y=0 leaves −ln(1−p). Confident wrong probabilities are therefore penalized heavily. Probabilities must be numerically clipped to avoid log(0); libraries implement such clipping. The Source: scikit-learn API Reference, log_loss.
The formula applies only when quantities are defined as in the source and background. Check boundary cases, measurement conditions and whether an approximation rather than equality is intended. Substitution can be algebraically correct yet technically wrong when assumptions fail or the population or system boundary changes. Validity limit: For each example, y=1 leaves −ln(p), while y=0 leaves −ln(1−p). Confident wrong probabilities are therefore penalized heavily. Probabilities must be numerically clipped to avoid log(0); libraries implement such clipping. The Source: scikit-learn API Reference, log_loss.
The stated units are: nats per example with natural logarithms; dimensionless mean. Convert values to a consistent unit system before substitution; counts, percentages and dimensionless ratios are not interchangeable. The variable list states the role of each symbol. Dimensional analysis catches many errors but does not replace checking the formula's assumptions. Validity limit: For each example, y=1 leaves −ln(p), while y=0 leaves −ln(1−p). Confident wrong probabilities are therefore penalized heavily. Probabilities must be numerically clipped to avoid log(0); libraries implement such clipping. The Source: scikit-learn API Reference, log_loss.
Comparison helps when both quantities address the same question but emphasize different properties. Report the definition, input data, threshold or averaging with the value. The source scikit-learn API Reference, log_loss. supports the definition used here; other conventions can differ. Choose the measure to match the learning or analysis goal. Validity limit: For each example, y=1 leaves −ln(p), while y=0 leaves −ln(1−p). Confident wrong probabilities are therefore penalized heavily. Probabilities must be numerically clipped to avoid log(0); libraries implement such clipping. The Source: scikit-learn API Reference, log_loss.
Quanta exam set
Curated exam set for "Binary log loss":
Question (front)
Which formula describes Binary log loss?
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Question (front)
How do you rearrange Binary log loss for Defining relation?
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Which common mistake happens with Binary log loss?
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Create a curated FSRS exam set for L = mean binary cross-entropy: formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.
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