Compute the **Expected Calibration Error (ECE)** to measure how well a classifier’s predicted probabilities match actual outcomes. ECE bins predictions by confidence and compares average confidence to empirical accuracy in each bin.

The metric is defined as:
$$
\mathrm{ECE} = \sum_{b=1}^{B}\frac{|S_b|}{n}\left|\mathrm{acc}(S_b)-\mathrm{conf}(S_b)\right|
$$
where $S_b$ is the set of indices in bin $b$, $n$ is the total number of samples, $\mathrm{acc}(S_b)$ is average correctness, and $\mathrm{conf}(S_b)$ is average predicted confidence.

## Requirements

Implement the function:
```python
def expected_calibration_error(y_true, y_prob, n_bins=10):
    \"""
    Computes Expected Calibration Error (ECE) for binary classification.

    Args:
        y_true (np.ndarray): True labels (0 or 1), length n.
        y_prob (np.ndarray): Predicted probabilities for class 1, length n.
        n_bins (int): Number of equal-width confidence bins in [0, 1].

    Returns:
        float: The Expected Calibration Error (ECE).
    \"""
    # Your solution below
```

**Rules:**
- Use **equal-width bins** over the interval $[0, 1]$ (e.g., for 10 bins: [0.0,0.1), …, [0.9,1.0]).
- For each bin, compute accuracy as the fraction of samples with `y_true == (y_prob >= 0.5)` (i.e., predicted label uses threshold 0.5).
- For each bin, compute confidence as the mean of `y_prob` in that bin.
- Combine bins using the weighted sum shown in the formula (empty bins contribute 0).
- Use only **NumPy** and Python built-in libraries (no sklearn/scipy).

## Example

```python
y_true = [1, 0, 1, 1, 0]
y_prob = [0.9, 0.6, 0.2, 0.8, 0.1]

expected_calibration_error(y_true, y_prob, n_bins=5)
```

Output:
```python
0.28
```

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Kendall Tau Correlation

Calibration error

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73. Calibration error

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Similar Questions

Argument	Type
n_bins	int
y_prob	np.ndarray
y_true	np.ndarray