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Antigravity Agent
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---
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id: wiki-2026-0508-bayes-theorem
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title: Bayes' Theorem
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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: [Bayes Rule, Bayes Law, Conditional Probability Inversion]
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duplicate_of: none
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source_trust_level: A
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confidence_score: 0.98
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verification_status: applied
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tags: [probability, statistics, inference, mathematics, 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: SciPy / NumPy
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---
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# Bayes' Theorem
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## 매 한 줄
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> **"매 P(A|B) = P(B|A) × P(A) / P(B) — conditional probability 의 inversion 의 통한 evidence-based belief revision 의 mathematical foundation"**. Reverend Thomas Bayes (1763 posthumous) 의 essay, Laplace (1774) 의 generalize, 2026 modern ML 의 entire Bayesian stack — diffusion model 의 noise schedule, Kalman filter, LLM uncertainty calibration — 의 core.
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## 매 핵심
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### 매 공식 the form
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- **Standard**: `P(A|B) = P(B|A) × P(A) / P(B)`
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- **Odds form**: `O(A|B) = O(A) × LR` where `LR = P(B|A)/P(B|¬A)`
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- **Discrete partition**: `P(H_i|E) = P(E|H_i)P(H_i) / Σⱼ P(E|H_j)P(H_j)`
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- **Continuous**: `p(θ|D) = p(D|θ)p(θ) / ∫p(D|θ)p(θ)dθ`
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### 매 terminology
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- **Prior** P(A): pre-evidence belief
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- **Likelihood** P(B|A): evidence-given-hypothesis
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- **Posterior** P(A|B): post-evidence belief
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- **Evidence / Marginal** P(B): normalizing constant
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### 매 응용
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1. Medical testing — base-rate-aware diagnosis (mammography paradox).
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2. Spam filtering — Naive Bayes classifier.
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3. Search & rescue — posterior heatmap update from sensor sweep.
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4. LLM 의 token sampling — temperature-scaled posterior over vocabulary.
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## 💻 패턴
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### Medical test (base rate problem)
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```python
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def bayes_diagnosis(prevalence: float, sensitivity: float, specificity: float) -> dict:
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"""Disease prevalence 1%, test 99% sensitive + 95% specific.
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Positive test => actual disease probability?"""
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p_disease = prevalence
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p_pos_given_disease = sensitivity
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p_pos_given_healthy = 1 - specificity
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p_pos = p_pos_given_disease * p_disease + p_pos_given_healthy * (1 - p_disease)
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p_disease_given_pos = (p_pos_given_disease * p_disease) / p_pos
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return {
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"P(disease | +test)": p_disease_given_pos,
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"P(healthy | +test)": 1 - p_disease_given_pos,
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}
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print(bayes_diagnosis(0.01, 0.99, 0.95)) # ~16.6% — counter-intuitive
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```
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### Naive Bayes spam (log-space)
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```python
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import numpy as np
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from collections import Counter
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class NaiveBayesSpam:
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def __init__(self, alpha=1.0):
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self.alpha = alpha # Laplace smoothing
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def fit(self, docs, labels):
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self.classes = np.unique(labels)
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self.log_prior = {c: np.log((labels == c).mean()) for c in self.classes}
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self.vocab = set(w for d in docs for w in d.split())
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V = len(self.vocab)
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self.log_lik = {}
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for c in self.classes:
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words = Counter(w for d, l in zip(docs, labels) if l == c for w in d.split())
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total = sum(words.values()) + self.alpha * V
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self.log_lik[c] = {w: np.log((words.get(w, 0) + self.alpha) / total)
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for w in self.vocab}
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return self
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def predict(self, doc):
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scores = {c: self.log_prior[c] + sum(self.log_lik[c].get(w, 0)
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for w in doc.split())
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for c in self.classes}
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return max(scores, key=scores.get)
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```
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### Bayesian A/B (closed-form Beta-Binomial)
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```python
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from scipy import stats
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def prob_b_beats_a(a_clicks, a_imp, b_clicks, b_imp, n_samples=100_000):
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a = stats.beta(1 + a_clicks, 1 + a_imp - a_clicks).rvs(n_samples)
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b = stats.beta(1 + b_clicks, 1 + b_imp - b_clicks).rvs(n_samples)
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return (b > a).mean()
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print(f"P(B>A) = {prob_b_beats_a(73, 1000, 91, 1010):.3f}")
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```
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### Odds-form rapid update
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```python
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def odds_update(prior_odds: float, likelihood_ratio: float) -> float:
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"""Posterior odds = prior odds × LR. Mental-arithmetic friendly."""
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return prior_odds * likelihood_ratio
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# DNA match: prior 1:1000, LR = 100,000
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print(odds_update(1/1000, 100_000)) # 100 → P ≈ 99%
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```
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### Kalman filter (Bayesian, Gaussian)
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```python
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def kalman_step(mu, sigma2, z, R, Q):
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"""Predict + update; everything Bayesian under Normal-Normal conjugate."""
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# predict (process noise Q)
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sigma2 = sigma2 + Q
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# update (sensor z, sensor noise R)
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K = sigma2 / (sigma2 + R)
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mu = mu + K * (z - mu)
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sigma2 = (1 - K) * sigma2
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return mu, sigma2
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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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| Conjugate prior 의 fit | closed-form posterior |
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| Discrete + small | exact enumeration |
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| Continuous + nonconjugate | MCMC (NUTS / HMC) |
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| Streaming sensor data | Kalman / particle filter |
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| Class imbalance + features | Naive Bayes baseline |
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**기본값**: probabilistic classification 의 default — Naive Bayes (log-space) + Laplace smoothing.
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## 🔗 Graph
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- 부모: [[Statistical-Analysis]]
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- 변형: [[Bayesian-Updating]] · [[Belief-Revision]]
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- 응용: [[Item-Item-Collaborative-Filtering]] · [[몬테카를로 시뮬레이션]]
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- Adjacent: [[Inference-Coupled Persistence]] · [[Multi-agent-System]]
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## 🤖 LLM 활용
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**언제**: probabilistic reasoning 의 explanation, base-rate-aware decision, evidence weighting.
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**언제 X**: deterministic logic 의 sufficient 인 경우 — overhead 의 X.
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## ❌ 안티패턴
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- **Base-rate neglect**: P(B|A) 의 confuse with P(A|B) — prosecutor's fallacy.
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- **Naive equal prior**: domain knowledge 의 ignore 의 인해 prior 의 default uniform.
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- **Evidence double-counting**: dependent evidence 의 conditional independence 의 assume.
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- **Improper normalization**: continuous case 의 evidence integral 의 omit.
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## 🧪 검증 / 중복
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- Verified (Jaynes *Probability Theory: The Logic of Science*, Pearl *Causality* 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 Bayes' theorem with medical, NB, A/B, Kalman patterns |
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