2nd/10_Wiki/Topics/Other/Bayes-Theorem.md

id, title, category, status, canonical_id, aliases, duplicate_of, source_trust_level, confidence_score, verification_status, tags, raw_sources, last_reinforced, github_commit, tech_stack

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

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)

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)

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

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)

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