Fundamentals of 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)

$P( B|A) = \frac{P(B) P(A|B)}{P(A)}$

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

$P( B|A_1, A_2) = \frac{P(B|A_1) P(A_2|B, A_1)P(A_1)}{P(A_1, A_2)}$

(add $A_1$ to the end)
P(B| $A_1$ ): prior probability

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.

最后编辑于：2018.10.17 06:03:09