Monte Carlo Methods

🎯 Learning Objectives

• Understand the law of large numbers and its role in Monte Carlo
• Estimate integrals using Monte Carlo simulation
• Compute confidence intervals for Monte Carlo estimates
• Implement variance reduction techniques

1.1 Intuition: Estimating by Random Sampling

🔑 Key Insight: The law of large numbers guarantees that Monte Carlo estimates converge to the true value as sample size increases.

1.2 Monte Carlo Integration

python
Run Inline
import numpy as np
# estimate integral of sin(x) from 0 to pi
n = 10000
x = np.random.uniform(0, np.pi, n)
estimate = np.pi * np.mean(np.sin(x))
print(f"MC estimate: {estimate:.4f}, true: 2.0")

1.3 Variance Reduction

Antithetic Variates

python
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u = np.random.uniform(0, 1, n//2)
estimates = (f(u) + f(1-u)) / 2

Control Variates

1.4 Confidence Intervals

✅ Practice Questions

Solution

python
Run Inline
import numpy as np
x = np.random.uniform(0, 1, 10000)
estimate = np.mean(np.exp(x))  # since (b-a) = 1
print(estimate)  # approx 1.718 (true: e-1 = 1.71828)

Solution

MC error $\propto \sigma / \sqrt{n}$. To halve the error, quadruple $n$ (this is the curse of MC — slow convergence). Join Discord NextImportance Sampling