---
id: wiki-2026-0508-markov-chain-monte-carlo
title: Markov Chain Monte Carlo (MCMC)
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [MCMC, Metropolis-Hastings, Gibbs Sampling, HMC, NUTS]
duplicate_of: none
source_trust_level: A
confidence_score: 0.9
verification_status: applied
tags: [bayesian, sampling, mcmc, hmc, pymc, numpyro]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack: { language: Python, framework: PyMC/NumPyro }
---

# Markov Chain Monte Carlo (MCMC)

## 매 한 줄
> **"매 MCMC = stationary distribution이 target인 chain 만들기"**. 정규화 상수 모르고도 posterior 샘플 가능.

## 매 핵심
### 매 알고리즘
- **Metropolis-Hastings**: propose q(x'|x), accept α=min(1, π(x')q(x|x')/(π(x)q(x'|x))).
- **Random walk MH**: q = Normal(x, σ²). σ가 acceptance 결정.
- **Gibbs**: 조건부 p(x_i | x_{-i}) 순차 샘플. conjugate에 강함.
- **Slice sampling**: 보조 변수, tuning 적음.
- **HMC (Hamiltonian)**: gradient + leapfrog. high-dim 효율.
- **NUTS**: HMC trajectory 자동 결정. Stan/PyMC/NumPyro 기본.
- **SMC, parallel tempering**: multi-modal에 유리.

### 매 진단
- **Trace plot**: chain 안정성 시각 검사
- **R̂ (Gelman-Rubin)**: 다중 chain 수렴, <1.01 권장
- **ESS (effective sample size)**: 자기상관 보정 샘플 수
- **Energy diagnostic** (HMC): divergent transitions 0 목표
- **Posterior predictive check**: model fit

### 매 응용
1. Bayesian posterior 추정 (intractable normalizer)
2. Hierarchical models (multilevel regression)
3. Latent variable models
4. Bayesian deep learning (BNN, variational alternative)
5. Phylogenetics, epidemiology

## 💻 패턴
### Metropolis-Hastings (numpy)
```python
import numpy as np
def mh(log_target, x0, n=10000, step=0.5, rng=np.random.default_rng()):
    x = np.array(x0, dtype=float); samples = [x.copy()]
    log_p = log_target(x); accepts = 0
    for _ in range(n):
        x_new = x + rng.normal(scale=step, size=x.shape)
        log_p_new = log_target(x_new)
        if np.log(rng.uniform()) < log_p_new - log_p:
            x, log_p = x_new, log_p_new; accepts += 1
        samples.append(x.copy())
    return np.array(samples), accepts / n      # target ~0.234 (high-d), 0.44 (1d)
```

### Gibbs sampling (bivariate normal)
```python
def gibbs_bvn(rho, n=10000, rng=np.random.default_rng()):
    x = y = 0.0; out = np.empty((n, 2))
    for i in range(n):
        x = rng.normal(rho * y, np.sqrt(1 - rho**2))
        y = rng.normal(rho * x, np.sqrt(1 - rho**2))
        out[i] = (x, y)
    return out
```

### PyMC (NUTS)
```python
import pymc as pm, numpy as np
y = np.random.normal(2, 1, 100)
with pm.Model() as m:
    mu = pm.Normal("mu", 0, 10)
    sigma = pm.HalfNormal("sigma", 1)
    pm.Normal("y", mu, sigma, observed=y)
    idata = pm.sample(2000, tune=1000, chains=4, target_accept=0.9)
print(pm.summary(idata, var_names=["mu", "sigma"]))   # r_hat, ess
```

### NumPyro (JAX, fast)
```python
import jax, numpyro
import numpyro.distributions as dist
from numpyro.infer import MCMC, NUTS

def model(y):
    mu = numpyro.sample("mu", dist.Normal(0, 10))
    sigma = numpyro.sample("sigma", dist.HalfNormal(1))
    numpyro.sample("obs", dist.Normal(mu, sigma), obs=y)

mcmc = MCMC(NUTS(model), num_warmup=1000, num_samples=2000, num_chains=4)
mcmc.run(jax.random.PRNGKey(0), y=y)
mcmc.print_summary()
```

### Diagnostics (ArviZ)
```python
import arviz as az
az.plot_trace(idata)
az.plot_rank(idata)
print(az.rhat(idata).max(), az.ess(idata).min())
az.plot_pair(idata, divergences=True)
```

### 효율 팁
```python
# 1) Reparameterize (non-centered): theta = mu + sigma * z, z~N(0,1)
# 2) target_accept 0.9~0.99 if divergences
# 3) 표준화/스케일링 → leapfrog 안정
# 4) Initial values: pm.find_MAP() or jitter+adapt_diag
```

## 매 결정 기준
| 상황 | Sampler |
|---|---|
| Low-dim, custom posterior | MH (간단) |
| Conjugate hierarchical | Gibbs |
| Continuous, gradient 가능 | NUTS/HMC |
| Discrete latent | MH within Gibbs, SMC |
| Multi-modal | Parallel tempering, SMC |
| 대용량 / GPU | NumPyro (JAX), BlackJAX |
| 빠른 prod 근사 | VI (대안), Laplace |

**기본값**: continuous → NumPyro/PyMC NUTS. Discrete → Gibbs / SMC.

## 🔗 Graph
- 부모: [[Bayesian-Inference]]
- 변형: [[Metropolis-Hastings]], [[Gibbs-Sampling]], [[NUTS]]
- 응용: [[Bayesian-Regression]]
- Adjacent: [[Variational-Inference]], [[Stan]]

## 🤖 LLM 활용
**언제**: 모델 작성, sampler 선택, divergence 진단 가이드.
**언제 X**: 복잡 hierarchical model 검증은 도메인 전문가 + posterior predictive.

## ❌ 안티패턴
- Centered hierarchical에 NUTS 그대로 (divergences) → non-centered
- Single chain → R̂ 불가, 수렴 진단 X
- Burn-in/warmup 무시
- Acceptance rate 99% (step 너무 작음) or 1% (너무 큼)
- Trace plot 안 보고 mean만 신뢰
- VI로 충분한데 MCMC 돌리기 (시간 낭비)

## 🧪 검증 / 중복
- Verified (Gelman BDA3, Neal HMC review, Hoffman NUTS, PyMC/NumPyro docs). 신뢰도 A.
- 중복: 없음.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — PyMC/NumPyro 패턴, ArviZ diagnostic |