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Bayes Theorem

Theory

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Bayes Theorem

Bayes: P(H|E) = P(E|H) P(H) / P(E). Prior P(H), likelihood P(E|H), posterior P(H|E). Spam: H = spam, E = word “lottery”. A rare disease with a pretty-good test still has a modest posterior if the prior is tiny — classic trap.

Always name prior, likelihood, posterior. Plug tiny numbers if they ask to compute. Don’t recite the formula with no story.

Bayes Theorem — output — 0.8. After seeing lottery, spam belief jumps from 0.2 to 0.8.

Diagram

prior P(H)
       × P(E|H)
       / P(E)
       ▼
  posterior P(H|E)

Exam tip

Spam or disease story + name the three terms.

Example

# Bayes — lottery word
# P(spam)=0.2  P(lottery|spam)=0.8  P(lottery|ham)=0.05
# P(lottery)=0.8*0.2 + 0.05*0.8 = 0.20
p_spam_given_lottery = (0.8 * 0.2) / 0.20
print(round(p_spam_given_lottery, 2))

Bayes Theorem — output: 0.8. After seeing lottery, spam belief jumps from 0.2 to 0.8.

Short notes

P(H|E) = P(E|H)P(H)/P(E)
Prior / likelihood / posterior
Trapignoring the prior.

# Bayes — lottery word # P(spam)=0.2 P(lottery|spam)=0.8 P(lottery|ham)=0.05 # P(lottery)=0.8*0.2 + 0.05*0.8 = 0.20 p_spam_given_lottery = (0.8 * 0.2) / 0.20 print(round(p_spam_given_lottery, 2))

Bayes Theorem

Example

Short notes

Questions

Bayes Theorem

Example

Short notes

Questions