2nd/10_Wiki/Topic_General/From_Other/Bayesian-Updating.md

---
id: wiki-2026-0508-bayesian-updating
title: Bayesian Updating
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [Bayesian Inference, Posterior Update, Belief Updating]
duplicate_of: none
source_trust_level: A
confidence_score: 0.95
verification_status: applied
tags: [statistics, inference, probability, ml, decision-theory]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: Python
  framework: PyMC / NumPyro
---

# Bayesian Updating

## 매 한 줄
> **"매 Posterior ∝ Likelihood × Prior — evidence 의 arrival 마다 belief 의 incremental refinement"**. Bayes (1763) 의 sermon 에서 출발 의, 2026 modern stack 의 PyMC 5, NumPyro 0.15, Stan 2.34 의 통한 millions-of-parameters posterior 의 NUTS / HMC sampling 의 routine.

## 매 핵심

### 매 공식
- **Bayes' rule**: `P(H|E) = P(E|H) × P(H) / P(E)`
- **Sequential update**: `posterior_t = likelihood_t × posterior_{t-1}`
- **Log-form** (numerical stability): `log P(H|E) = log P(E|H) + log P(H) - log P(E)`

### 매 conjugate priors
- Beta–Binomial (CTR, conversion rate)
- Gamma–Poisson (event counts, arrival rate)
- Normal–Normal (sensor fusion, A/B continuous metric)
- Dirichlet–Multinomial (categorical preferences)

### 매 응용
1. A/B testing — early-stopping, peeking 의 robust handling.
2. Spam filter — Naive Bayes 의 incremental email update.
3. Robot localization — particle filter 의 prior 와 sensor likelihood 의 fuse.
4. LLM uncertainty — token-level posterior 의 calibration (2026 Anthropic constitutional classifiers).

## 💻 패턴

### Beta–Binomial conjugate (CTR)
```python
from scipy import stats
import numpy as np

# Prior: Beta(1, 1) = uniform
alpha, beta = 1.0, 1.0

# Observe: 73 clicks out of 1000 impressions
clicks, impressions = 73, 1000
alpha_post = alpha + clicks
beta_post  = beta + (impressions - clicks)

posterior = stats.beta(alpha_post, beta_post)
print(f"Posterior mean CTR: {posterior.mean():.4f}")
print(f"95% credible interval: {posterior.interval(0.95)}")
```

### Sequential update (online)
```python
def online_beta_update(alpha, beta, click: bool):
    return (alpha + click, beta + (1 - click))

a, b = 1.0, 1.0
for event in stream_of_clicks():
    a, b = online_beta_update(a, b, event)
    if a + b > 100:  # confident enough
        decide(stats.beta(a, b).mean())
```

### PyMC 5 hierarchical
```python
import pymc as pm
import numpy as np

variants = ["A", "B", "C"]
clicks = np.array([73, 91, 82])
impressions = np.array([1000, 1010, 990])

with pm.Model() as model:
    mu = pm.Beta("mu", 1, 1)
    kappa = pm.HalfNormal("kappa", 10)
    theta = pm.Beta("theta", mu * kappa, (1 - mu) * kappa, shape=len(variants))
    pm.Binomial("y", n=impressions, p=theta, observed=clicks)
    idata = pm.sample(2000, tune=1000, target_accept=0.95)

pm.summary(idata, var_names=["theta"])
```

### NumPyro NUTS (GPU-accelerated, JAX)
```python
import numpyro
import numpyro.distributions as dist
from numpyro.infer import MCMC, NUTS
import jax.numpy as jnp

def model(impressions, clicks=None):
    p = numpyro.sample("p", dist.Beta(1, 1))
    numpyro.sample("obs", dist.Binomial(impressions, p), obs=clicks)

mcmc = MCMC(NUTS(model), num_warmup=500, num_samples=2000)
mcmc.run(jax.random.PRNGKey(0), impressions=jnp.array(1000), clicks=jnp.array(73))
mcmc.print_summary()
```

### Bayesian online change-point detection
```python
def bocpd_step(observation, run_length_probs, hazard=1/250):
    """Adams & MacKay 2007."""
    pred = compute_predictive_prob(observation, run_length_probs)
    growth = run_length_probs * pred * (1 - hazard)
    cp = (run_length_probs * pred * hazard).sum()
    new = np.concatenate([[cp], growth])
    return new / new.sum()
```

## 매 결정 기준
| 상황 | Approach |
|---|---|
| 작은 N + conjugate prior 의 fit | closed-form (Beta–Binomial) |
| Hierarchical + ~10k params | PyMC NUTS (CPU) |
| Large model + GPU 의 가능 | NumPyro (JAX) |
| Streaming / sub-ms latency | Online conjugate update |
| Discrete latent 의 dominant | particle filter / variational |

**기본값**: A/B test 의 default — Beta–Binomial conjugate + 95% credible interval.

## 🔗 Graph
- 부모: [[Bayes-Theorem]]
- 변형: [[Belief-Revision]] · [[Inference-Coupled Persistence]]
- 응용: [[Item-Item-Collaborative-Filtering]] · [[Statistical-Analysis]]
- Adjacent: [[몬테카를로 시뮬레이션]] · [[Multi-agent-System]]

## 🤖 LLM 활용
**언제**: A/B early-stopping decision, sensor fusion, parameter uncertainty 의 explicit propagation.
**언제 X**: data 의 abundant + flat likelihood 의 dominant 인 경우 — frequentist MLE 의 sufficient.

## ❌ 안티패턴
- **Improper prior 의 use**: posterior 의 not normalize 의 가능 — proper prior 의 verify.
- **Prior 의 sneaking strong assumption**: subjective prior 의 sensitivity analysis 의 필수.
- **Peeking 의 misinterpretation**: Bayesian posterior 의 frequentist p-value 의 X — separate calibration.
- **MCMC convergence 의 무시**: R-hat > 1.01, ESS < 400 의 즉시 의 reject.

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

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — full Bayesian updating with PyMC 5, NumPyro, online BOCPD |