---
id: wiki-2026-0508-bayes-theorem
title: Bayes' Theorem
category: 10_Wiki/Topics
status: verified
canonical_id: self
aliases: [Bayes Rule, Bayes Law, Conditional Probability Inversion]
duplicate_of: none
source_trust_level: A
confidence_score: 0.98
verification_status: applied
tags: [probability, statistics, inference, mathematics, decision-theory]
raw_sources: []
last_reinforced: 2026-05-10
github_commit: pending
tech_stack:
  language: Python
  framework: SciPy / NumPy
---

# Bayes' Theorem

## 매 한 줄
> **"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.

## 매 핵심

### 매 공식 the form
- **Standard**: `P(A|B) = P(B|A) × P(A) / P(B)`
- **Odds form**: `O(A|B) = O(A) × LR` where `LR = P(B|A)/P(B|¬A)`
- **Discrete partition**: `P(H_i|E) = P(E|H_i)P(H_i) / Σⱼ P(E|H_j)P(H_j)`
- **Continuous**: `p(θ|D) = p(D|θ)p(θ) / ∫p(D|θ)p(θ)dθ`

### 매 terminology
- **Prior** P(A): pre-evidence belief
- **Likelihood** P(B|A): evidence-given-hypothesis
- **Posterior** P(A|B): post-evidence belief
- **Evidence / Marginal** P(B): normalizing constant

### 매 응용
1. Medical testing — base-rate-aware diagnosis (mammography paradox).
2. Spam filtering — Naive Bayes classifier.
3. Search & rescue — posterior heatmap update from sensor sweep.
4. LLM 의 token sampling — temperature-scaled posterior over vocabulary.

## 💻 패턴

### Medical test (base rate problem)
```python
def bayes_diagnosis(prevalence: float, sensitivity: float, specificity: float) -> dict:
    """Disease prevalence 1%, test 99% sensitive + 95% specific.
       Positive test => actual disease probability?"""
    p_disease = prevalence
    p_pos_given_disease = sensitivity
    p_pos_given_healthy = 1 - specificity
    
    p_pos = p_pos_given_disease * p_disease + p_pos_given_healthy * (1 - p_disease)
    p_disease_given_pos = (p_pos_given_disease * p_disease) / p_pos
    
    return {
        "P(disease | +test)": p_disease_given_pos,
        "P(healthy | +test)": 1 - p_disease_given_pos,
    }

print(bayes_diagnosis(0.01, 0.99, 0.95))  # ~16.6% — counter-intuitive
```

### Naive Bayes spam (log-space)
```python
import numpy as np
from collections import Counter

class NaiveBayesSpam:
    def __init__(self, alpha=1.0):
        self.alpha = alpha  # Laplace smoothing
    
    def fit(self, docs, labels):
        self.classes = np.unique(labels)
        self.log_prior = {c: np.log((labels == c).mean()) for c in self.classes}
        self.vocab = set(w for d in docs for w in d.split())
        V = len(self.vocab)
        self.log_lik = {}
        for c in self.classes:
            words = Counter(w for d, l in zip(docs, labels) if l == c for w in d.split())
            total = sum(words.values()) + self.alpha * V
            self.log_lik[c] = {w: np.log((words.get(w, 0) + self.alpha) / total)
                                for w in self.vocab}
        return self
    
    def predict(self, doc):
        scores = {c: self.log_prior[c] + sum(self.log_lik[c].get(w, 0)
                                               for w in doc.split())
                  for c in self.classes}
        return max(scores, key=scores.get)
```

### Bayesian A/B (closed-form Beta-Binomial)
```python
from scipy import stats

def prob_b_beats_a(a_clicks, a_imp, b_clicks, b_imp, n_samples=100_000):
    a = stats.beta(1 + a_clicks, 1 + a_imp - a_clicks).rvs(n_samples)
    b = stats.beta(1 + b_clicks, 1 + b_imp - b_clicks).rvs(n_samples)
    return (b > a).mean()

print(f"P(B>A) = {prob_b_beats_a(73, 1000, 91, 1010):.3f}")
```

### Odds-form rapid update
```python
def odds_update(prior_odds: float, likelihood_ratio: float) -> float:
    """Posterior odds = prior odds × LR. Mental-arithmetic friendly."""
    return prior_odds * likelihood_ratio

# DNA match: prior 1:1000, LR = 100,000
print(odds_update(1/1000, 100_000))  # 100 → P ≈ 99%
```

### Kalman filter (Bayesian, Gaussian)
```python
def kalman_step(mu, sigma2, z, R, Q):
    """Predict + update; everything Bayesian under Normal-Normal conjugate."""
    # predict (process noise Q)
    sigma2 = sigma2 + Q
    # update (sensor z, sensor noise R)
    K = sigma2 / (sigma2 + R)
    mu = mu + K * (z - mu)
    sigma2 = (1 - K) * sigma2
    return mu, sigma2
```

## 매 결정 기준
| 상황 | Approach |
|---|---|
| Conjugate prior 의 fit | closed-form posterior |
| Discrete + small | exact enumeration |
| Continuous + nonconjugate | MCMC (NUTS / HMC) |
| Streaming sensor data | Kalman / particle filter |
| Class imbalance + features | Naive Bayes baseline |

**기본값**: probabilistic classification 의 default — Naive Bayes (log-space) + Laplace smoothing.

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

## 🤖 LLM 활용
**언제**: probabilistic reasoning 의 explanation, base-rate-aware decision, evidence weighting.
**언제 X**: deterministic logic 의 sufficient 인 경우 — overhead 의 X.

## ❌ 안티패턴
- **Base-rate neglect**: P(B|A) 의 confuse with P(A|B) — prosecutor's fallacy.
- **Naive equal prior**: domain knowledge 의 ignore 의 인해 prior 의 default uniform.
- **Evidence double-counting**: dependent evidence 의 conditional independence 의 assume.
- **Improper normalization**: continuous case 의 evidence integral 의 omit.

## 🧪 검증 / 중복
- Verified (Jaynes *Probability Theory: The Logic of Science*, Pearl *Causality* 2nd).
- 신뢰도 A.

## 🕓 Changelog
| 날짜 | 변경 |
|---|---|
| 2026-05-08 | Phase 1 |
| 2026-05-10 | Manual cleanup — full Bayes' theorem with medical, NB, A/B, Kalman patterns |