#P03 · 概率統計與AI優化 (Probability & AI) · Lv.3在3D思維導圖中查看

貝氏定理 (後驗概率推斷)

Thomas Bayes & Pierre-Simon Laplace · 1763년

P(H \mid E) = \frac{P(E \mid H)P(H)}{P(E)} = \frac{P(E \mid H)P(H)}{P(E\mid H)P(H)+P(E\mid H^c)P(H^c)}

\text{Posterior} \propto \text{Likelihood} \times \text{Prior}

結合先驗概率 P(H) 與新證據 E 的似然度 P(E|H) 來更新後驗概率 P(H|E)，是統計推斷與AI不確定性推理的核心定律。

🧮 即時公式計算器與圖形繪製器

先驗概率 P(H)

靈敏度 P(E|H)

假陽性率 P(E|Hᶜ)

P(H\mid E) = \frac{0.95\times 0.05}{0.1425} = 33.33\%

사후확률 P(H|E)

33.33%

2차 연속 양성 시 사후확률

82.61%

전체 양성률 P(E)

14.25%

先修概念路徑 (Prerequisite Path) (5 STEP)

按順序點擊即可沿3D霓虹路徑從基礎直達目標公式。

ax + b = 0 \implies x = -\frac{b}{a}

(a+b)^2 = a^2 + 2ab + b^2

{}_nP_r = \frac{n!}{(n-r)!}

P(A \mid B) = \frac{P(A \cap B)}{P(B)}

P(H \mid E) = \frac{P(E \mid H)P(H)}{P(E)} = \frac{P(E \mid H)P(H)}{P(E\mid H)P(H)+P(E\mid H^c)P(H^c)}

'원인→결과' 확률 P(증상|질병)만 알고 있을 때, 거꾸로 '결과→원인' 확률 P(질병|증상)을 희귀도(사전확률)까지 반영해 정확히 뒤집어 줍니다.

直接驅動垃圾郵件過濾(樸素貝氏)、醫療AI診斷校準、自動駕駛卡爾曼/粒子濾波及貝氏超參數優化。

#Spam Filter#Bayesian Inference#Medical AI Diagnosis#Kalman Filter

const bayes = (prior, sens, fpr) => (sens * prior) / (sens * prior + fpr * (1 - prior));