---
id: wiki-2026-0508-monte-carlo-integration
title: Monte Carlo Integration
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [Monte-Carlo-Integration, MC-Integration, 몬테카를로-적분]
duplicate_of: none
source_trust_level: A
confidence_score: 0.95
verification_status: applied
tags: [numerical, integration, sampling, statistics, simulation]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: python
  framework: numpy-jax
---

# Monte Carlo Integration

## 매 한 줄
> **"매 무작위 샘플의 평균이 적분값으로 수렴"**. ∫f dμ ≈ (1/N)Σf(xᵢ), error O(N⁻¹/²) — 매 dimension에 무관한 게 매 핵심 강점이다. 1949 Metropolis-Ulam에서 Manhattan Project 이후, 2026 LLM 시대에도 매 RLHF reward estimation·diffusion sampling의 backbone.

## 매 핵심

### 매 estimator
- **Standard MC**: x ~ p, Î = (1/N)Σf(xᵢ); Var = σ²/N.
- **Importance sampling**: x ~ q, Î = (1/N)Σf(xᵢ)p(xᵢ)/q(xᵢ).
- **Control variates**: f → f − c(g − E[g]); 매 variance ↓.
- **Stratified**: 매 domain partition.
- **Quasi-MC**: Sobol/Halton — error O(N⁻¹ logᵈ N).

### 매 수렴
- Error std ~ σ/√N (CLT).
- 매 dim 무관 — high-dim integration의 매 유일한 실용 도구.

### 매 응용
1. Bayesian inference — posterior expectation (MCMC).
2. Computer graphics — path tracing, light transport.
3. Finance — option pricing (Black-Scholes path).
4. RLHF — reward expectation.
5. Diffusion model — score-matching expectation.

## 💻 패턴

### Basic MC integral
```python
import numpy as np
def mc_integrate(f, low, high, n=10000):
    x = np.random.uniform(low, high, n)
    return (high - low) * f(x).mean(), (high - low) * f(x).std() / np.sqrt(n)
```

### Importance sampling
```python
def importance_mc(f, sampler_q, log_p, log_q, n=10000):
    x = sampler_q(n)
    w = np.exp(log_p(x) - log_q(x))
    return (f(x) * w).mean()
```

### Control variates
```python
def cv_mc(f, g, Eg, n=10000):
    x = np.random.uniform(0, 1, n)
    fx, gx = f(x), g(x)
    c = -np.cov(fx, gx)[0, 1] / np.var(gx)
    return (fx + c * (gx - Eg)).mean()
```

### Quasi-MC with Sobol
```python
from scipy.stats.qmc import Sobol
sampler = Sobol(d=5, scramble=True)
points = sampler.random_base2(m=14)  # 2^14 points
estimate = f(points).mean()
```

### MCMC (Metropolis-Hastings)
```python
def mh(log_pi, x0, n=10000, sigma=0.5):
    x, samples = x0, [x0]
    for _ in range(n):
        x_prop = x + sigma * np.random.randn(*x.shape)
        if np.log(np.random.rand()) < log_pi(x_prop) - log_pi(x):
            x = x_prop
        samples.append(x)
    return np.array(samples)
```

### JAX vectorized MC
```python
import jax, jax.numpy as jnp
@jax.jit
def mc(key, n):
    x = jax.random.uniform(key, (n,))
    return jnp.mean(jnp.exp(-x**2))
```

## 매 결정 기준
| 상황 | Method |
|---|---|
| Smooth low-dim | Quadrature or QMC |
| High-dim | Vanilla MC |
| Heavy tail / rare event | Importance sampling |
| Posterior | MCMC (NUTS, HMC) |
| Light transport | Path tracing + MIS |

**기본값**: Vanilla MC + control variates (low complexity, low variance).

## 🔗 Graph
- 부모: [[Statistics]]
- 변형: [[MCMC]]
- 응용: [[Bayesian Inference]] · [[RLHF]]

## 🤖 LLM 활용
**언제**: High-dim integration, expectation under intractable distribution, simulation.
**언제 X**: 1-3 dim smooth functions (use Gauss quadrature).

## ❌ 안티패턴
- **Variance 무시**: 매 std error 안 보고 estimate 제출.
- **Bad importance proposal**: 매 q tail이 p보다 얇으면 explosion.
- **Correlated samples**: MCMC autocorrelation 무시 → 매 ESS 부풀려짐.

## 🧪 검증 / 중복
- Verified (Robert & Casella "Monte Carlo Statistical Methods").
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — MC variants + JAX/MCMC patterns |