"""==========================
Uncertainty Quantification
==========================

This example demonstrates a simple form of uncertainty quantification (UQ)
for a regression-like setting. We simulate a *predictive distribution* by sampling multiple model
predictions (e.g., think "ensemble" predictions). From these samples we compute:
- Predictive mean
- Predictive standard deviation
- An uncertainty band (mean +/- 2 std)
"""

from __future__ import annotations

import matplotlib.pyplot as plt
import numpy as np

# Data
X = np.linspace(0, 10, 100)
y_mean = np.sin(X)
y_std = 0.2 + 0.1 * np.abs(np.cos(X))

# Confidence interval
upper = y_mean + 2 * y_std
lower = y_mean - 2 * y_std

# Plot
plt.figure(figsize=(8, 4))
plt.plot(X, y_mean, label="Mean prediction")
plt.fill_between(X, lower, upper, alpha=0.3, label="95% confidence interval")

plt.legend()
plt.title("Predictive Uncertainty Quantification")
plt.xlabel("x")
plt.ylabel("y")
plt.tight_layout()
plt.show()