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Predicting Probabilities

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Dr. Rishabh Pathak

1. What is Predicting Probabilities?

  • The predict_proba method estimates the probability that a given input belongs to each class.
  • It returns values in the range [0, 1], representing the model's confidence as probabilities.
  • The sum of predicted probabilities across all classes for a sample is always 1 (i.e., they form a valid probability distribution).

2. Output Shape of predict_proba

  • For binary classification, the shape of the output is (n_samples, 2):
  • Column 0: Probability of the sample belonging to the negative class.
  • Column 1: Probability of the sample belonging to the positive class.
  • For multiclass classification, the shape is (n_samples, n_classes), with each column corresponding to the probability of the sample belonging to that class.

3. Interpretation of predict_proba Output

  • The probability reflects how confidently the model believes a data point belongs to each class.
  • For example, in binary classification:

Input Sample

Predicted Probability (Negative Class)

Predicted Probability (Positive Class)

1

0.2

0.8

2

0.9

0.1
  • The model predicts positive class if the positive class probability is greater than a threshold (default 0.5).

4. Relation to Thresholding and Classification

  • The default threshold for making classification decisions is 0.5:
  • If predict_proba for positive class > 0.5, sample is classified as positive.
  • Otherwise, it is classified as negative.
  • You can adjust this threshold depending on the problem, which affects false positive and false negative rates.
  • Adjusting thresholds can optimize metrics like precision, recall, F-score, especially on imbalanced datasets.

5. Calibration of Probability Estimates

  • Not all models produce well-calibrated probabilities.
  • A calibrated model outputs probabilities that closely match true likelihoods.
  • Example of a poor calibration: a decision tree grown to full depth might assign probability 1 or 0, but be often wrong.
  • Calibration can be improved using methods like:
  • Platt scaling
  • Isotonic regression
  • Reference: Paper by Niculescu-Mizil and Caruana, “Predicting Good Probabilities with Supervised Learning”.

6. Examples Using predict_proba (from the book)

  • Using a GradientBoostingClassifier on toy datasets:
# Suppose gbrt is a trained GradientBoostingClassifier
print("Shape of probabilities:", gbrt.predict_proba(X_test).shape)
# Output:
# Shape of probabilities: (n_samples, 2)
 
print("Predicted probabilities:\n", gbrt.predict_proba(X_test[:6]))
  • Output shows actual predicted probabilities for each class:
[[0.1 0.9]
[0.8 0.2]
[0.7 0.3]
...
]
  • The first column corresponds to the first class probability, the second column to the second class.

7. Advantages of predict_proba

  • Provides interpretable uncertainty estimates in terms of probabilities.
  • Useful for decision making where probabilistic thresholds are preferable to hard decisions.
  • Can be integrated into pipelines that weigh risks (e.g., medical diagnosis, fraud detection).
  • Helps in ranking samples by probability to prioritize further analysis.

8. Relationship Between predict_proba and decision_function

  • Some classifiers implement both decision_function and predict_proba:
  • decision_function returns raw scores or margins.
  • predict_proba converts these scores to probabilities.
  • Probabilities are usually obtained by applying a logistic function or softmax on the decision function scores.
  • Calibrated models provide better probability estimates compared to raw scores alone,.

9. Practical Considerations

  • When probabilities are needed (e.g., for risk assessment), prefer models supporting predict_proba.
  • Be cautious that probabilities are only as good as model calibration.
  • Always validate probabilities with calibration plots or metrics like Brier score.

10. Summary Table

Aspect

Details

Purpose

Provides class membership probabilities

Output Shape

Binary: (n_samples, 2), Multiclass: (n_samples, n_classes)

Values

Probabilities between 0 and 1, sum to 1 per sample

Default threshold

0.5 for binary classification

Calibration

Models may need calibration for accurate probabilities

Applications

Threshold tuning, risk assessment, ranking predictions

Relation

Derived from decision_function scores via logistic or softmax

Example Models

GradientBoostingClassifier, Logistic Regression, Random Forest

 

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