Antigravity Agent
159 lines
5.3 KiB
Markdown
159 lines
5.3 KiB
Markdown
---
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id: wiki-2026-0508-bayesian-updating
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title: Bayesian Updating
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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: [Bayesian Inference, Posterior Update, Belief Updating]
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duplicate_of: none
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source_trust_level: A
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confidence_score: 0.95
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verification_status: applied
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tags: [statistics, inference, probability, ml, decision-theory]
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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:
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language: Python
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framework: PyMC / NumPyro
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---
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# Bayesian Updating
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## 매 한 줄
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> **"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.
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## 매 핵심
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### 매 공식
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- **Bayes' rule**: `P(H|E) = P(E|H) × P(H) / P(E)`
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- **Sequential update**: `posterior_t = likelihood_t × posterior_{t-1}`
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- **Log-form** (numerical stability): `log P(H|E) = log P(E|H) + log P(H) - log P(E)`
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### 매 conjugate priors
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- Beta–Binomial (CTR, conversion rate)
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- Gamma–Poisson (event counts, arrival rate)
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- Normal–Normal (sensor fusion, A/B continuous metric)
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- Dirichlet–Multinomial (categorical preferences)
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### 매 응용
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1. A/B testing — early-stopping, peeking 의 robust handling.
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2. Spam filter — Naive Bayes 의 incremental email update.
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3. Robot localization — particle filter 의 prior 와 sensor likelihood 의 fuse.
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4. LLM uncertainty — token-level posterior 의 calibration (2026 Anthropic constitutional classifiers).
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## 💻 패턴
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### Beta–Binomial conjugate (CTR)
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```python
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from scipy import stats
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import numpy as np
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# Prior: Beta(1, 1) = uniform
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alpha, beta = 1.0, 1.0
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# Observe: 73 clicks out of 1000 impressions
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clicks, impressions = 73, 1000
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alpha_post = alpha + clicks
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beta_post = beta + (impressions - clicks)
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posterior = stats.beta(alpha_post, beta_post)
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print(f"Posterior mean CTR: {posterior.mean():.4f}")
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print(f"95% credible interval: {posterior.interval(0.95)}")
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```
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### Sequential update (online)
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```python
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def online_beta_update(alpha, beta, click: bool):
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return (alpha + click, beta + (1 - click))
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a, b = 1.0, 1.0
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for event in stream_of_clicks():
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a, b = online_beta_update(a, b, event)
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if a + b > 100: # confident enough
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decide(stats.beta(a, b).mean())
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```
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### PyMC 5 hierarchical
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```python
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import pymc as pm
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import numpy as np
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variants = ["A", "B", "C"]
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clicks = np.array([73, 91, 82])
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impressions = np.array([1000, 1010, 990])
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with pm.Model() as model:
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mu = pm.Beta("mu", 1, 1)
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kappa = pm.HalfNormal("kappa", 10)
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theta = pm.Beta("theta", mu * kappa, (1 - mu) * kappa, shape=len(variants))
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pm.Binomial("y", n=impressions, p=theta, observed=clicks)
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idata = pm.sample(2000, tune=1000, target_accept=0.95)
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pm.summary(idata, var_names=["theta"])
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```
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### NumPyro NUTS (GPU-accelerated, JAX)
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```python
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import numpyro
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import numpyro.distributions as dist
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from numpyro.infer import MCMC, NUTS
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import jax.numpy as jnp
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def model(impressions, clicks=None):
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p = numpyro.sample("p", dist.Beta(1, 1))
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numpyro.sample("obs", dist.Binomial(impressions, p), obs=clicks)
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mcmc = MCMC(NUTS(model), num_warmup=500, num_samples=2000)
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mcmc.run(jax.random.PRNGKey(0), impressions=jnp.array(1000), clicks=jnp.array(73))
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mcmc.print_summary()
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```
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### Bayesian online change-point detection
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```python
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def bocpd_step(observation, run_length_probs, hazard=1/250):
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"""Adams & MacKay 2007."""
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pred = compute_predictive_prob(observation, run_length_probs)
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growth = run_length_probs * pred * (1 - hazard)
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cp = (run_length_probs * pred * hazard).sum()
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new = np.concatenate([[cp], growth])
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return new / new.sum()
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```
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## 매 결정 기준
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| 상황 | Approach |
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|---|---|
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| 작은 N + conjugate prior 의 fit | closed-form (Beta–Binomial) |
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| Hierarchical + ~10k params | PyMC NUTS (CPU) |
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| Large model + GPU 의 가능 | NumPyro (JAX) |
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| Streaming / sub-ms latency | Online conjugate update |
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| Discrete latent 의 dominant | particle filter / variational |
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**기본값**: A/B test 의 default — Beta–Binomial conjugate + 95% credible interval.
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## 🔗 Graph
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- 부모: [[Bayes-Theorem]]
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- 변형: [[Belief-Revision]] · [[Inference-Coupled Persistence]]
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- 응용: [[Item-Item-Collaborative-Filtering]] · [[Statistical-Analysis]]
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- Adjacent: [[몬테카를로 시뮬레이션]] · [[Multi-agent-System]]
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## 🤖 LLM 활용
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**언제**: A/B early-stopping decision, sensor fusion, parameter uncertainty 의 explicit propagation.
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**언제 X**: data 의 abundant + flat likelihood 의 dominant 인 경우 — frequentist MLE 의 sufficient.
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## ❌ 안티패턴
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- **Improper prior 의 use**: posterior 의 not normalize 의 가능 — proper prior 의 verify.
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- **Prior 의 sneaking strong assumption**: subjective prior 의 sensitivity analysis 의 필수.
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- **Peeking 의 misinterpretation**: Bayesian posterior 의 frequentist p-value 의 X — separate calibration.
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- **MCMC convergence 의 무시**: R-hat > 1.01, ESS < 400 의 즉시 의 reject.
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## 🧪 검증 / 중복
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- Verified (Gelman et al. *Bayesian Data Analysis* 3rd, McElreath *Statistical Rethinking* 2nd).
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- 신뢰도 A.
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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 — full Bayesian updating with PyMC 5, NumPyro, online BOCPD |
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