2nd/10_Wiki/Topics/AI_and_ML/Bayesian Statistics.md

---
id: wiki-2026-0508-bayesian-statistics
title: Bayesian Statistics
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [베이지안 통계, Bayes' theorem, posterior, prior, MCMC, variational inference, PyMC, Stan]
duplicate_of: none
source_trust_level: A
confidence_score: 0.95
verification_status: applied
tags: [bayesian, statistics, mcmc, variational-inference, pymc, stan, probabilistic-programming, uncertainty]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: Python
  framework: PyMC / Stan / NumPyro / Pyro
---

# Bayesian Statistics

## 📌 한 줄 통찰
> **"매 probability = 매 belief 의 degree"**. 매 frequency X — 매 prior + data → posterior 의 update. 매 small data + prior knowledge 의 strong. 매 result = 매 distribution (not point). 매 modern compute (MCMC / VI) 의 mainstream.

## 📖 핵심

### Bayes' theorem
$$P(\theta | D) = \frac{P(D | \theta) \cdot P(\theta)}{P(D)}$$

- **P(θ)**: prior — 매 belief.
- **P(D | θ)**: likelihood — 매 data 의 model.
- **P(θ | D)**: posterior — 매 update 된 belief.
- **P(D)**: evidence (normalizer).

### vs Frequentist
| 측면 | Frequentist | Bayesian |
|---|---|---|
| Probability | 매 long-run frequency | 매 belief degree |
| Parameter | 매 fixed unknown | 매 random variable |
| Result | 매 point + CI | 매 posterior distribution |
| Small data | 매 fragile | 매 prior 의 robust |
| Compute | 매 cheap | 매 expensive (until MCMC) |
| Interpretation | "95% of intervals contain θ" | "P(θ ∈ [a,b]) = 0.95" |

### 매 conjugate prior (analytical)
| Likelihood | Prior | Posterior |
|---|---|---|
| Binomial | Beta | Beta |
| Poisson | Gamma | Gamma |
| Normal (known σ) | Normal | Normal |
| Normal (unknown μ,σ) | Normal-Gamma | Normal-Gamma |
| Multinomial | Dirichlet | Dirichlet |

→ 매 closed-form 가, 매 limited.

### 매 inference (modern)

#### MCMC (Markov Chain Monte Carlo)
- **Metropolis-Hastings**: 매 random walk + accept/reject.
- **Hamiltonian MC (HMC)**: 매 gradient 활용.
- **NUTS** (No-U-Turn): 매 HMC 의 auto-tune.
- ✅ 매 정확. ❌ 매 slow.

#### Variational Inference (VI)
- 매 posterior 의 approximate distribution q(θ) 의 fit.
- 매 KL divergence 의 minimize.
- ✅ 매 fast + scale. ❌ 매 approximate.

#### Sequential Monte Carlo
- 매 particle filter.
- 매 streaming OK.

### 매 응용
1. **A/B testing**: 매 frequentist 보다 매 interpretable.
2. **Hyperparameter tuning** (Bayesian Optimization): 매 GP + acquisition.
3. **Hierarchical models**: 매 group-level prior.
4. **Time series** (state-space): 매 Kalman, 매 particle filter.
5. **Causal inference** (Bayesian network): 매 DAG.
6. **Drug discovery / clinical**: 매 small N + strong prior.
7. **Robotics** (SLAM): 매 pose + map 의 joint.
8. **Topic modeling** (LDA): 매 Dirichlet prior.

### 매 modern stack
- **Stan**: 매 NUTS, 매 mature.
- **PyMC** (3 → 4 → 5): 매 Python + Aesara.
- **NumPyro**: 매 JAX-based, 매 fast.
- **Pyro**: 매 PyTorch + VI.
- **TFP**: 매 TensorFlow Probability.
- **Edward2 / blackjax**: 매 modular.

## 💻 패턴

### Coin flip (PyMC)
```python
import pymc as pm
import numpy as np

# 매 data: 매 8 head, 매 2 tail
data = np.array([1]*8 + [0]*2)

with pm.Model() as model:
    p = pm.Beta('p', alpha=2, beta=2)  # 매 prior
    obs = pm.Bernoulli('obs', p=p, observed=data)
    trace = pm.sample(2000, return_inferencedata=True)

# 매 posterior
import arviz as az
az.plot_posterior(trace)
print(az.summary(trace))
# p mean ≈ 0.71, hdi_3% ≈ 0.50, hdi_97% ≈ 0.89
```

### Hierarchical (group-level)
```python
with pm.Model() as h:
    # 매 hyperprior
    mu = pm.Normal('mu', 0, 10)
    sigma = pm.HalfNormal('sigma', 5)
    
    # 매 group-level
    theta = pm.Normal('theta', mu, sigma, shape=n_groups)
    
    # 매 likelihood
    y = pm.Normal('y', theta[group_idx], 1, observed=data)
    
    trace = pm.sample(2000)
```

→ 매 partial pooling — 매 group 의 small N 의 borrow strength.

### Bayesian A/B test
```python
with pm.Model() as ab:
    p_a = pm.Beta('p_a', 1, 1)
    p_b = pm.Beta('p_b', 1, 1)
    
    obs_a = pm.Binomial('obs_a', n=n_a, p=p_a, observed=conv_a)
    obs_b = pm.Binomial('obs_b', n=n_b, p=p_b, observed=conv_b)
    
    diff = pm.Deterministic('diff', p_b - p_a)
    
    trace = pm.sample(2000)

# 매 P(B > A)
prob_b_better = (trace.posterior['diff'] > 0).mean().item()
print(f'P(B > A) = {prob_b_better:.3f}')
```

→ 매 frequentist 보다 매 actionable.

### Variational inference (faster)
```python
import numpyro
import numpyro.distributions as dist
from numpyro.infer import SVI, Trace_ELBO
from numpyro.infer.autoguide import AutoNormal

def model(data):
    p = numpyro.sample('p', dist.Beta(2, 2))
    numpyro.sample('obs', dist.Bernoulli(p), obs=data)

guide = AutoNormal(model)
svi = SVI(model, guide, optim.Adam(0.01), Trace_ELBO())
state = svi.init(jax.random.PRNGKey(0), data)
for step in range(2000):
    state, loss = svi.update(state, data)
```

### Bayesian Optimization (hyperparameter)
```python
from skopt import gp_minimize
from skopt.space import Real, Integer

def objective(params):
    lr, depth = params
    return train_and_eval(lr, depth)  # 매 minimize

result = gp_minimize(
    objective,
    [Real(1e-5, 1e-1, prior='log-uniform', name='lr'),
     Integer(1, 10, name='depth')],
    n_calls=50,
)
```

### Posterior predictive check
```python
with model:
    ppc = pm.sample_posterior_predictive(trace)

# 매 simulated data 의 actual 의 비교 — 매 model fit 의 visual.
az.plot_ppc(az.from_pymc3(posterior_predictive=ppc, model=model))
```

## 🤔 결정 기준
| 상황 | Method |
|---|---|
| Small data + prior | Conjugate (analytical) |
| Complex model + accuracy | NUTS (PyMC / Stan) |
| Large data + speed | VI (Pyro / NumPyro) |
| Streaming | Particle filter |
| Hyperparameter tune | BO (skopt / Optuna) |
| A/B test | Beta-Binomial + Bayes |
| Topic modeling | LDA |
| Causal | Bayesian network |

**기본값**: PyMC + NUTS 의 baseline. 매 scale 가 NumPyro / VI.

## 🔗 Graph
- 부모: [[Statistics]] · [[Probability-Theory]]
- 변형: [[MCMC]] · [[Variational-Inference]] · [[Bayesian-Network]]
- 응용: [[Bayesian-Optimization]] · [[LDA]] · [[SLAM]]
- Tool: [[PyMC]] · [[Stan]]
- Adjacent: [[Bayes-Theorem]] · [[Bayesian-Updating]]

## 🤖 LLM 활용
**언제**: 매 small data + prior. 매 uncertainty quantify. 매 hierarchical structure. 매 hyperparameter tune.
**언제 X**: 매 large data + speed > accuracy. 매 simple frequentist 의 OK.

## ❌ 안티패턴
- **Improper prior**: 매 posterior 의 invalid.
- **No PPC**: 매 fit 의 모름.
- **MCMC 의 chains 1**: 매 convergence 의 detect X.
- **Burn-in 무시**: 매 biased estimate.
- **Conjugate 의 force**: 매 wrong likelihood.
- **VI 의 over-confident** (mean-field): 매 underestimate uncertainty.
- **R-hat ignore**: 매 non-convergence.

## 🧪 검증 / 중복
- Verified (Gelman BDA, McElreath Statistical Rethinking, Stan/PyMC docs).
- 신뢰도 A.
- Related: [[Bayes-Theorem]] · [[MCMC]] · [[Bayesian-Optimization]] · [[Variational-Inference]].

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — Bayes formula + MCMC / VI + 매 PyMC / NumPyro / skopt code |