Semantic & Factorization

Student Example

##     I  D  G   Prob
## 1  i0 d0 g1  0.126
## 2  i0 d0 g2  0.168
## 3  i0 d0 g3  0.126
## 4  i0 d1 g1  0.009
## 5  i0 d1 g2  0.045
## 6  i0 d1 g3  0.126
## 7  i1 d0 g1  0.252
## 8  i1 d0 g2 0.0224
## 9  i1 d0 g3 0.0056
## 10 i1 d1 g1   0.06
## 11 i1 d1 g2  0.036
## 12 i1 d1 g3  0.024
  • Grade \(G = \{g^1 (A), g^2 (B), g^3 (C)\}\)
  • Course Dsifficulty \(D = \{d^0 (easy), d^1 (hard)\}\)
  • Student Intelligence \(I = \{i^0 (low), i^1 (hight)\}\)
  • Student SAT \(S\)
  • Reference Letter \(L\)

Let think about construct the dependency of these attributes

  • Grade depends on Diffulculty and Intelligence
  • SAT depends on Intelligence
  • Letter depends on Grade

If a student’s intelligence (\(i^0\)), and he takes a easy course (\(d^0\)), the probability he get grade A is very high \(0.9\)

If a stduent’s intelligence (\(i^1\)), the probability he get low SAT score (\(s^0\)) is very low \(0.2\)

If a student have grade A (\(g^1\)), the probability this student have a good reference letter (\(I^1\)) is very high \(0.9\)

\[P(G,D,I,S,L)\]

Question 1

Answer for questions

Question 1

Answer

\(P(D)P(I)P(G|I,D)P(S|I)P(L|G)\)

Explaination

We can directly apply the chain rule for Bayesian networks here.

This comes from the standard chain rule of probability, which states that

\[P(D,I,G,S,L)=P(D)P(I|D)P(G|D,I)P(S|D,I,G)P(L|D,I,G,S).\]

We can then apply the conditional independencies in the graph to simplify this equation.