Answer for Qn V, Part E

[Subbarao Kambhampati <rao@asu.edu>]

Some people have been clamoring for the answer to the Qn V Part E (which I 
had changed in specs, but the solutions given still reflect the old question).
The idea seems to be to keep me as busy as you guys are... :-(

Anyways, here is the answer to the version that I asked you to do. I will 
link this up from the online solutions.


Rao
Answer to Qn V Part E

We just found that there the slurpees at Apus's place have
liquefied. We also found out, from reliable sources, that the batch of
heavy water they are using at the reactor is INDEED low quality
one. Compute the probability distribution on Homer glowing in the
dark. Use the enumeration method as discussed on Friday's
class to compute this probability (notice that this involves
marginalization and normalization). [If you want to be sure, You can
check your answer by plugging this network in the bayesnet tool and
doing the query).


Answer:

We are trying to find out P(GD|SL,LH)


P(GD|SL,LH) = P(GD,SL,LH)/P(SL,LH)

rather than compute P(SL,LH), we will use normalization and let alpha=
1/P(SL,LH) 
(the way we do in enumeration). Thus

P(GD|SL,LH) = alpha * P(GD,SL,LH)

P(~GD|SL,LH) = alpha * P(~GD,SL,LH)



Computing P(GD,SL,LH), we need to marginalize over CM, IP

P(GD,SL,LH) = .Sum.over.CM=cm .Sum.over.IP=ip  P(GD,SL,LH,cm,ip)
      
  = P(GD,SL,LH,IP,CM) 0.03780
   +P(GD,SL,LH,IP,~CM) 0.000180
   +P(GD,SL,LH,~IP,~CM) 0.00098
   +P(GD,SL,LH,~IP,CM)  0.03780

  = 0.07676


[[ Explanation:
Since each element of the sum above is a joint probability table enty,
using the bayes net semantics, we can compute each of these entries
exactly as shown. 

For example, P(GD,SL,LH,~IP,~CM) = P(GD|~CM) * P(SL|~CM) *P(CM|LH,~IP) * P(LH) * P(~IP)
  
                                 = 0.05      * 0.1       *  0.3        * 0.4   * 0.7

                                 = 0.000420
Similarly
 P(GD,SL,LH,~IP,~CM) = P(GD|~CM)[.05]  * P(SL|~CM)[.1] * P(~CM|~IP,LH)[1-0.3] * P(~IP)[1-.3] *P(LH)[.4]
             = .05*.1*.7*.7*.4 = 0.00098

and so on..
]] 

Similarly, we can compute

P(~GD,SL,LH) as the sum of entries

  = P(~GD,SL,LH,IP,CM) 0.03780
   +P(~GD,SL,LH,IP,~CM) 0.003420
   +P(~GD,SL,LH,~IP,~CM) 0.018620
   +P(~GD,SL,LH,~IP,CM)  0.03780
 
 = 0.09764
[Notice that the first and fourth entries are the same as before..this
is because P(~GD|CM) = P(GD|CM) =0.5]]


So

P(GD|SL,LH)  = alpha * 0.07676
P(~GD|SL,LH) = alpha * 0.09764

Summing both sides, we get

1 = alpha* (0.07676+0.09764)

alpha = 5.733944

Plugging it in the equation for P(GD|SL,LH)

we get

P(GD|SL,LH) = 5.733944*0.07676=0.44013 (which is what you will get if
you used the bayes net tool!)

So, the probability that Homer glows in the dark given slurpee
liquefaction and low quality heavy water is ~0.44

[[If you use the bayes tool, you will find that that 
Probability of glowing in the dark was 0.15 when you have no
evidence. Becomes 0.24 if LW alone is found, 0.376 if SL alone is
found, and .44 if both are found]]