---
id: wiki-2026-0508-bayesian-inference
title: Bayesian Inference
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [Bayesian Inference, Bayesian Statistics, Posterior Inference]
duplicate_of: none
source_trust_level: A
confidence_score: 0.95
verification_status: applied
tags: [statistics, ml, probabilistic, mcmc]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: python
  framework: pymc,numpyro,stan
---

# Bayesian Inference

## 매 한 줄
> **"매 prior + likelihood = posterior — 매 belief 의 evidence 에 의해 의 update"**. Bayes 1763 의 origin, 20세기 frequentist 의 dominance, 2026 의 NumPyro/PyMC + GPU MCMC + variational inference 의 mainstream — 매 LLM uncertainty quantification 의 backbone.

## 매 핵심

### 매 Bayes rule
P(θ|D) = P(D|θ) P(θ) / P(D)
- **P(θ)**: prior — 매 data 이전 의 belief.
- **P(D|θ)**: likelihood — 매 model 의 data fit.
- **P(θ|D)**: posterior — updated belief.
- **P(D)**: evidence (marginal likelihood).

### 매 4 inference 방법
- **Conjugate**: closed-form (Beta-Bernoulli · Gaussian-Gaussian).
- **MCMC**: HMC · NUTS · Gibbs — exact (asymptotic) · slow.
- **Variational (VI)**: posterior 의 simpler family 의 approximate — fast · biased.
- **Sequential MC**: particle filter — 매 streaming · state-space.

### 매 응용
1. A/B test (Bayesian alternative 의 frequentist).
2. Hierarchical model (분류 multi-level).
3. ML calibration · uncertainty (BNN · Gaussian process).
4. LLM logit calibration · RAG confidence.

## 💻 패턴

### NumPyro: Bayesian linear regression (NUTS)
```python
import numpyro
import numpyro.distributions as dist
from numpyro.infer import NUTS, MCMC
import jax.numpy as jnp
import jax.random as random

def model(X, y=None):
    alpha = numpyro.sample("alpha", dist.Normal(0., 10.))
    beta  = numpyro.sample("beta",  dist.Normal(jnp.zeros(X.shape[1]), 1.))
    sigma = numpyro.sample("sigma", dist.HalfNormal(1.))
    mu    = alpha + X @ beta
    numpyro.sample("obs", dist.Normal(mu, sigma), obs=y)

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

### PyMC: Bayesian A/B test (Beta-Bernoulli)
```python
import pymc as pm

with pm.Model():
    p_a = pm.Beta("p_a", alpha=1, beta=1)
    p_b = pm.Beta("p_b", alpha=1, beta=1)
    pm.Binomial("obs_a", n=n_a, p=p_a, observed=conv_a)
    pm.Binomial("obs_b", n=n_b, p=p_b, observed=conv_b)
    diff = pm.Deterministic("lift", p_b - p_a)
    idata = pm.sample(2000, tune=1000, chains=4)

prob_b_better = (idata.posterior["lift"] > 0).mean().item()
```

### Hierarchical model (varying intercept)
```python
def hierarchical(X, group_idx, y=None, n_groups=10):
    mu_a    = numpyro.sample("mu_a",    dist.Normal(0., 5.))
    sigma_a = numpyro.sample("sigma_a", dist.HalfNormal(1.))
    a       = numpyro.sample("a", dist.Normal(mu_a, sigma_a).expand([n_groups]))
    beta    = numpyro.sample("beta", dist.Normal(0., 1.))
    sigma   = numpyro.sample("sigma", dist.HalfNormal(1.))
    mu      = a[group_idx] + beta * X
    numpyro.sample("obs", dist.Normal(mu, sigma), obs=y)
```

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

guide = AutoNormal(model)
svi   = SVI(model, guide, numpyro.optim.Adam(1e-3), Trace_ELBO())
state = svi.init(random.PRNGKey(0), X, y)

for i in range(2000):
    state, loss = svi.update(state, X, y)
params = svi.get_params(state)
```

### Conjugate update (Beta-Bernoulli online)
```python
class BetaBernoulli:
    def __init__(self, alpha=1, beta=1):
        self.alpha, self.beta = alpha, beta

    def update(self, success: bool):
        self.alpha += int(success)
        self.beta  += int(not success)

    def mean(self):  return self.alpha / (self.alpha + self.beta)
    def credible_interval(self, q=0.95):
        from scipy.stats import beta
        return beta.interval(q, self.alpha, self.beta)
```

### Bayesian neural net (Pyro Bayesian layer)
```python
import torch
import pyro
import pyro.nn as pnn

class BNN(pnn.PyroModule):
    def __init__(self, in_d, out_d):
        super().__init__()
        self.linear = pnn.PyroModule[torch.nn.Linear](in_d, out_d)
        self.linear.weight = pnn.PyroSample(dist.Normal(0., 1.).expand([out_d, in_d]).to_event(2))
        self.linear.bias   = pnn.PyroSample(dist.Normal(0., 1.).expand([out_d]).to_event(1))

    def forward(self, x, y=None):
        mean = self.linear(x).squeeze(-1)
        with pyro.plate("data", x.shape[0]):
            pyro.sample("obs", dist.Normal(mean, 0.1), obs=y)
        return mean
```

## 매 결정 기준
| 상황 | Method |
|---|---|
| Conjugate model · streaming | Conjugate update |
| 매 small model · accurate posterior | NUTS/HMC |
| Large data · fast approx | SVI · ADVI |
| State-space · time-series | Particle filter |
| Deep model · scale | BNN + variational · MC dropout |
| 매 hyperparameter optimization | Gaussian process + acquisition |

**기본값**: NumPyro + NUTS — 매 GPU/JAX 의 fast.

## 🔗 Graph
- 부모: [[Probability Theory]] · [[Statistical Inference]]
- 변형: [[MCMC]] · [[Variational Inference]] · [[Empirical Bayes]]
- 응용: [[Belief-System]] · [[Bayesian A/B Test]] · [[Bayesian Neural Network]]
- Adjacent: [[Frequentist Inference]] · [[Causal Inference]]

## 🤖 LLM 활용
**언제**: model 의 prior · likelihood 의 spec 의 draft, posterior plot 의 interpret, NumPyro/PyMC code 의 generate.
**언제 X**: 매 convergence diagnostic (R-hat · ESS · trace plot) — LLM 의 statistical judgment 의 unreliable, statistician 의 review 의 require.

## ❌ 안티패턴
- **Flat prior 의 always**: weak data + flat prior → unstable posterior. Weakly informative prior 의 use.
- **No convergence check**: R-hat > 1.01 · ESS < 400 → posterior 의 invalid.
- **Single chain MCMC**: multi-chain 의 mix check 의 mandatory.
- **Posterior point-estimate 의 only**: 매 distribution 의 entirety 의 use — credible interval · posterior predictive.

## 🧪 검증 / 중복
- Verified (Gelman *Bayesian Data Analysis* 3rd, NumPyro/PyMC/Stan docs, McElreath *Statistical Rethinking* 2nd).
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — Bayes rule + 4 methods + NumPyro/PyMC patterns |