Bias–variance decomposition helps you explain *why* a machine learning algorithm makes prediction errors by splitting expected error into parts from underfitting and overfitting. In this question, you’ll compute bias, variance, and noise from repeated model predictions on the same inputs.

## Requirements

Implement the function
```python
import numpy as np

def bias_variance_decomposition(y_true, predictions):
    \"""
    Computes bias, variance, and noise (irreducible error) using repeated predictions.

    Bias-variance decomposition (per input x) for squared loss:
        Let f_hat_j(x) be the prediction from run/model j.
        Mean prediction: mu(x) = (1/m) * sum_j f_hat_j(x)
        Bias^2: (mu(x) - y(x))^2
        Variance: (1/m) * sum_j (f_hat_j(x) - mu(x))^2
        Noise: (1/m) * sum_j (f_hat_j(x) - y(x))^2 - Bias^2 - Variance

    Overall values are the averages over all inputs x.

    Args:
        y_true (np.ndarray): True targets of length n.
        predictions (np.ndarray): 2D array of shape (m, n), where predictions[j][i]
            is the prediction for input i from run/model j.

    Returns:
        tuple[float, float, float]: (bias_squared, variance, noise)
    \"""
    # Your solution below
```

**Rules:**
- Use the squared error decomposition: $$\mathbb{E}[(\hat{f}(x)-y)^2] = \text{Bias}(\hat{f}(x))^2 + \text{Var}(\hat{f}(x)) + \sigma^2.$$
- Treat each row in `predictions` as an independent run (e.g., different training set, bootstrap sample, or random seed) of the same `ml_algorithm`.
- Return **global** `bias_squared`, `variance`, and `noise` as averages across all `n` inputs.
- Use only **NumPy** (and Python built-ins); don’t use any ML libraries.

## Example

```python
y_true = [3.0, 5.0]
predictions = [
    [2.0, 6.0],
    [4.0, 4.0],
    [3.0, 5.0],
]

bias2, var, noise = bias_variance_decomposition(y_true, predictions)
(bias2, var, noise)
```

Output:
```python
(0.0, 0.6666666667, 0.0)
```

Reduce Array Size to The Half

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Bias variance decomposition

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148. Bias variance decomposition

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Similar Questions

Argument	Type
y_true	np.ndarray
predictions	np.ndarray