#P03 · Probability, Statistics & AI · Lv.3View in 3D Mind-Map

Bayes' Theorem

Thomas Bayes & Pierre-Simon Laplace · 1763 CE

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}

Inverts conditional probabilities to update the prior belief P(H) into the posterior probability P(H|E) upon observing new evidence E.

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Prior P(H)

Sensitivity P(E|H)

False Positive 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%

Prerequisite Path (5 STEP)

Click steps in order to follow the 3D neon path from basics to target.

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.

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