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The question is asking which of the listed options is **not** a desirable property of the cross-entropy loss \( L \) for logistic regression. Let's analyze each of the options in the context of cross-entropy loss.

### Cross-Entropy Loss
In logistic regression, the cross-entropy loss \( L \) is defined as:

\[
L = -\left[ y \log(p) + (1-y) \log(1-p) \right]
\]

Where:
- \( y \) is the true label (0 or 1),
- \( p \) is the predicted probability of the positive class (output of the logistic function).

### Properties of Cross-Entropy Loss:
1. **High loss for misclassification**:
   - If the model predicts the wrong class (misclassification), the loss should be high to penalize the model. This is a desirable property.
   
2. **Low loss for correct classification**:
   - If the model predicts the correct class (low error), the loss should be low, rewarding the model. This is a desirable property.
   
3. **Consistency \( L = 0 \) if \( p = y \)**:
   - When the predicted probability exactly matches the true label, the loss should be zero. This is desirable, as it indicates perfect prediction.

4. **Non-negativity \( L \geq 0 \)**:
   - The loss should always be non-negative, as negative loss would not have a meaningful interpretation in this context.

### Analysis of the Options:
- **Option (a)**: " \( L \) should be very high for misclassification"

  - This is a desirable property, as discussed.

- **Option (b)**: " \( L \) should be very low for misclassification"

  - This is **not** a desirable property because if the model misclassifies, we want the loss to be high, not low. This seems to be the correct answer to the question.

- **Option (c)**: " \( L = 0 \) if \( p = y \)"

  - This is a desirable property, indicating no error when the prediction is perfect.

- **Option (d)**: "Consistency \( L \geq 0 \)"

  - This is a desirable property, ensuring the loss is always non-negative.

### Conclusion:
The correct answer is **Option (b)**: 

\[ L \) should be very low for misclassification \]

Would you like further details on any of these concepts? Here are some related questions you might consider:

1. How is the cross-entropy loss derived from the concept of entropy in information theory?
2. What is the relationship between cross-entropy loss and the likelihood function in logistic regression?
3. How does cross-entropy loss behave when the model is very confident but wrong?
4. Can cross-entropy loss be negative, and if not, why?
5. How does cross-entropy loss differ from mean squared error in the context of classification?

**Tip:** Understanding why high loss for misclassification is crucial can help you design better objective functions for machine learning models.

Explore the properties of cross-entropy loss in logistic regression. Learn why high loss for misclassification is crucial and how it ensures model performance. Suitable for advanced learners and practitioners.

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