title: Baysian Network Calculation of P(L1|i0) from Kollar Student Network
body: <p>In Daphne Koller's Student Network Model:
  <a href="http://spark-university.s3.amazonaws.com/stanford-pgm/slides/2.1.2-Repn-BNs-patterns.pdf" rel="nofollow noreferrer">The pdf in the link below: Reasoning Patterns, gives the graphical representation of the Student Baysian Network</a></p>
  
  <p>The problem I'm having is to calculate P(L1 | i0), that is the probability the student gets a high letter of recommendation L1 given he's not that smart i0. Since L1 has no conditional distribution with intelligence i, but both i and L have conditional distributions with G. The Joint Distribution is:</p>
  
  <pre><code>P(L1 i0 G) = P(L1 | G)P(G | i0)P(i0)
  </code></pre>
  
  <p>Since L depends on i0 through G: 
      P(L1 | G)P(G | i0) = P(L1|i0)
  Also, L and i0 have no conditional distribution so:
      P(L1 i0 G) -> P(L1 G)P(i0 G) = 0.5x0.5 = 0.25
  This gives:
      P(L1|i0) = P(L1 i0 G)/P(i0)
  From the model:
      P(i0) = 0.7
  This gives P(L1 | i0) = 0.25/0.7 = 0.357
  The answer in the Link above is 0.39</p>
  
  <p>How do you get that answer?</p>