Discrete Probability2

Bayes’ Theorem

• Bayes’ Theorem
  • 
  • p(E)≠ 0 and p(F) ≠ 0
• Generalized Bayes’ Theorem
  • 𝑝(𝐹j|𝐸)= 𝑝(𝐸|𝐹j)𝑝(𝐹j) / ∑𝑝(𝐸|𝐹i)𝑝(𝐹i)
  • p(E)≠ 0 for i = 1, 2, …, n
• Bayesian Spam Filters
  • B: set of spam messages
  • G: set of non-spam messages
  • w: particular word
  • nB(w), nG(w): the number of messages that it occurs in B and G
  • p(w): w가 포함된 스팸 메시지의 확률
    • p(w) = nB(w) / |B|
  • q(w): w가 포함된 스팸이 아닌 메시지의 확률
    • q(w) = nG(w) / |G|
  • S: the event that the message is spam
  • E: the event that the message contains w
  • p(S|E) = p(E|S)p(S) / p(E|S)p(S) + p(E|S-bar)p(S-bar)
    p(S|E) = p(E|S) / p(E|S) + p(E|S-bar) (p(S)=p(S-bar) = 1/2)
    r(w) = p(w) / (p(w) + q(w))
• Bayesian Spam Filters using Multiple Words