Fundamentals of Probability

Conditional Probability:

• Given B is ture, probability of A is true: P(A|B)
• Chain Rule:
  P(A, B): Both A and B are true.
  • P(A, B) = P(A|B) * P(B)
  • P(A, B, C) = P(A| B, C) * P(B| C) * P(C)

Bayes' theorem:

P(B): prior probobility.
We use it to test the probability firstly. Whether this method is better or worse depends.

(add to the end)
P(B| ): prior probability

Naive Bayes Classifier:

• Question: how to calculate P(C|F1, F2, ..., Fn)?
• Example: Fraud User Detection
  Prior information: Real User C0 = 77%; Fake User C1 = 23%
  Features:
  F1: Number of blogs -> s, m, l
  F2: Number of friends -> s, m, l
  F3: Real Avator or Not

  • Lemma:

    P(C| F1, F2, ..., Fn) = P(F1|C) * P(F2|C) * ... P(Fn|C) * P(C)/Z

    P(C| F1, F2, ..., Fn)
    = P(F1, F2, ..., Fn|C)*P(C)/P(F1, F2, ..., Fn)
    = P(F1, F2, ..., Fn|C)*P(C)/Z
    Because F1-Fn are all independent:
    P(F1, F2, ..., Fn|C) = P(F1|C) * P(F2|C) * ... P(Fn|C)

  • Complete Fraud User Detection:
    P(C0| F1 = S, F2 = M, F3 = R)
    = P(C0) *P(F1 = S| C0) *P(F2 = M| C0) *P(F3 = R| C0)/Z
    = 0.0623 / Z

• Conclusion: to calculate P(C|F1, F2, ..., Fn), we first calculate P(F1, F2, ...,Fn, C) to get the prior probability.