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이름만 다른(표기 변형) [[위키링크]]를 대상 문서의 canonical 제목으로 치환해 끊겼던 1,200개 링크를 연결. 제목/파일명 정규화 일치만 적용하고 별칭 매칭은 과병합 위험으로 제외(애매성 가드). 원본은 _link_reconcile_backup/ 에 백업. 도구: Datacollect/scripts/link_reconcile_apply.mjs Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
160 lines
5.1 KiB
Markdown
160 lines
5.1 KiB
Markdown
---
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id: wiki-2026-0508-markov-chain-monte-carlo
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title: Markov Chain Monte Carlo (MCMC)
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category: 10_Wiki/Topics
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status: verified
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canonical_id: self
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aliases: [MCMC, Metropolis-Hastings, Gibbs Sampling, HMC, NUTS]
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duplicate_of: none
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source_trust_level: A
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confidence_score: 0.9
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verification_status: applied
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tags: [bayesian, sampling, mcmc, hmc, pymc, numpyro]
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raw_sources: []
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last_reinforced: 2026-05-10
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github_commit: pending
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tech_stack: { language: Python, framework: PyMC/NumPyro }
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---
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# Markov Chain Monte Carlo (MCMC)
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## 매 한 줄
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> **"매 MCMC = stationary distribution이 target인 chain 만들기"**. 정규화 상수 모르고도 posterior 샘플 가능.
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## 매 핵심
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### 매 알고리즘
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- **Metropolis-Hastings**: propose q(x'|x), accept α=min(1, π(x')q(x|x')/(π(x)q(x'|x))).
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- **Random walk MH**: q = Normal(x, σ²). σ가 acceptance 결정.
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- **Gibbs**: 조건부 p(x_i | x_{-i}) 순차 샘플. conjugate에 강함.
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- **Slice sampling**: 보조 변수, tuning 적음.
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- **HMC (Hamiltonian)**: gradient + leapfrog. high-dim 효율.
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- **NUTS**: HMC trajectory 자동 결정. Stan/PyMC/NumPyro 기본.
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- **SMC, parallel tempering**: multi-modal에 유리.
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### 매 진단
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- **Trace plot**: chain 안정성 시각 검사
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- **R̂ (Gelman-Rubin)**: 다중 chain 수렴, <1.01 권장
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- **ESS (effective sample size)**: 자기상관 보정 샘플 수
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- **Energy diagnostic** (HMC): divergent transitions 0 목표
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- **Posterior predictive check**: model fit
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### 매 응용
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1. Bayesian posterior 추정 (intractable normalizer)
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2. Hierarchical models (multilevel regression)
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3. Latent variable models
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4. Bayesian deep learning (BNN, variational alternative)
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5. Phylogenetics, epidemiology
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## 💻 패턴
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### Metropolis-Hastings (numpy)
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```python
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import numpy as np
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def mh(log_target, x0, n=10000, step=0.5, rng=np.random.default_rng()):
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x = np.array(x0, dtype=float); samples = [x.copy()]
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log_p = log_target(x); accepts = 0
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for _ in range(n):
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x_new = x + rng.normal(scale=step, size=x.shape)
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log_p_new = log_target(x_new)
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if np.log(rng.uniform()) < log_p_new - log_p:
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x, log_p = x_new, log_p_new; accepts += 1
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samples.append(x.copy())
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return np.array(samples), accepts / n # target ~0.234 (high-d), 0.44 (1d)
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```
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### Gibbs sampling (bivariate normal)
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```python
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def gibbs_bvn(rho, n=10000, rng=np.random.default_rng()):
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x = y = 0.0; out = np.empty((n, 2))
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for i in range(n):
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x = rng.normal(rho * y, np.sqrt(1 - rho**2))
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y = rng.normal(rho * x, np.sqrt(1 - rho**2))
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out[i] = (x, y)
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return out
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```
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### PyMC (NUTS)
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```python
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import pymc as pm, numpy as np
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y = np.random.normal(2, 1, 100)
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with pm.Model() as m:
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mu = pm.Normal("mu", 0, 10)
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sigma = pm.HalfNormal("sigma", 1)
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pm.Normal("y", mu, sigma, observed=y)
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idata = pm.sample(2000, tune=1000, chains=4, target_accept=0.9)
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print(pm.summary(idata, var_names=["mu", "sigma"])) # r_hat, ess
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```
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### NumPyro (JAX, fast)
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```python
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import jax, numpyro
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import numpyro.distributions as dist
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from numpyro.infer import MCMC, NUTS
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def model(y):
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mu = numpyro.sample("mu", dist.Normal(0, 10))
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sigma = numpyro.sample("sigma", dist.HalfNormal(1))
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numpyro.sample("obs", dist.Normal(mu, sigma), obs=y)
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mcmc = MCMC(NUTS(model), num_warmup=1000, num_samples=2000, num_chains=4)
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mcmc.run(jax.random.PRNGKey(0), y=y)
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mcmc.print_summary()
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```
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### Diagnostics (ArviZ)
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```python
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import arviz as az
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az.plot_trace(idata)
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az.plot_rank(idata)
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print(az.rhat(idata).max(), az.ess(idata).min())
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az.plot_pair(idata, divergences=True)
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```
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### 효율 팁
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```python
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# 1) Reparameterize (non-centered): theta = mu + sigma * z, z~N(0,1)
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# 2) target_accept 0.9~0.99 if divergences
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# 3) 표준화/스케일링 → leapfrog 안정
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# 4) Initial values: pm.find_MAP() or jitter+adapt_diag
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```
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## 매 결정 기준
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| 상황 | Sampler |
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|---|---|
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| Low-dim, custom posterior | MH (간단) |
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| Conjugate hierarchical | Gibbs |
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| Continuous, gradient 가능 | NUTS/HMC |
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| Discrete latent | MH within Gibbs, SMC |
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| Multi-modal | Parallel tempering, SMC |
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| 대용량 / GPU | NumPyro (JAX), BlackJAX |
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| 빠른 prod 근사 | VI (대안), Laplace |
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**기본값**: continuous → NumPyro/PyMC NUTS. Discrete → Gibbs / SMC.
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## 🔗 Graph
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- 부모: [[Bayesian Inference]]
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- 변형: [[Metropolis-Hastings]], [[Gibbs-Sampling]], [[NUTS]]
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- 응용: [[Bayesian-Regression]]
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- Adjacent: [[Variational-Inference]], [[Stan]]
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## 🤖 LLM 활용
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**언제**: 모델 작성, sampler 선택, divergence 진단 가이드.
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**언제 X**: 복잡 hierarchical model 검증은 도메인 전문가 + posterior predictive.
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## ❌ 안티패턴
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- Centered hierarchical에 NUTS 그대로 (divergences) → non-centered
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- Single chain → R̂ 불가, 수렴 진단 X
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- Burn-in/warmup 무시
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- Acceptance rate 99% (step 너무 작음) or 1% (너무 큼)
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- Trace plot 안 보고 mean만 신뢰
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- VI로 충분한데 MCMC 돌리기 (시간 낭비)
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## 🧪 검증 / 중복
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- Verified (Gelman BDA3, Neal HMC review, Hoffman NUTS, PyMC/NumPyro docs). 신뢰도 A.
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- 중복: 없음.
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## 🕓 Changelog
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| 날짜 | 변경 |
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| 2026-05-08 | Phase 1 |
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| 2026-05-10 | Manual cleanup — PyMC/NumPyro 패턴, ArviZ diagnostic |
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