THE CONFIDENCE-CALIBRATION BIN

Difficulty: Medium Tags: Model Evaluation, Calibration, Statistics

Story

A risk model's calibration dashboard groups predicted probabilities into equal-width bins. Boundary handling must be consistent: each bin includes its left edge and excludes its right edge, except the final bin, which also includes 1.

The Math

For K bins and probability p in [0, 1], assign:

b(p) = min(K - 1, floor(K * p))

For each nonempty bin, compute the mean predicted probability and the empirical positive rate. For empty bins, output 0 for both.

Input Format

probs (length n), labels (length n, 0/1), K.

Output Format

K rows of (count, mean_prob, positive_rate), metrics to 6 decimal places.

Constraints

• 1 <= n <= 10^5, 1 <= K <= 1000, p in [0, 1], labels binary.
• Time limit: 1.0 second.

Example

Input

probs = [0.0, 0.5, 1.0], labels = [0, 1, 1], K = 2

Output

1 0.000000 0.000000
2 0.750000 1.000000

Explanation: Probability 0.5 belongs to the second bin, and probability 1.0 is explicitly clamped into the final bin.