Extending AnsProlog to reason with probabilities

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Extending AnsProlog to reason with probabilities

• Chitta Baral
• Arizona State university (joint work with Michael
  Gelfond, and Nelson Rushton)

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Motivation The Monty Hall problem

• A player is given the opportunity to select one
  of three closed doors, behind one of which there
  is a prize, and the other 2 rooms are empty.
• Once the player has made a selection, Monty is
  obligated to open one of the remaining closed
  doors, revealing that it does not contain the
  prize.
• Monty gives a choice to the player to switch to
  the other unopened door if he wants.
• Question Does it matter if the player switches,
  or
• Which unopened door has the higher probability of
  containing the prize?

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Go to White board

• http//www.remote.org/frederik/projects/ziege/bewe
  is.html

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Illustration-1

• First, let us assume that the car is behind door
  no. 1.
• We can do this without reducing the validity of
  our proof, because if the car were behind door
  no. 2, we only had to exchange all occurrences of
  "door 1" with "door 2" and vice versa, and the
  proof would still hold.
• The candidate has three choices of doors.
• Because he has no additional information, he
  randomly selects one.
• The possibility to choose each of the doors 1,
  2, or 3 is 1/3 each
•   Candidate chooses p  
• Door 1 1/3
• Door 2 1/3
• Door 3 1/3
• Sum 1 

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Illustration-2

• Going on from this table, we have to split the
  case depending on the door opened by the host.
• Since we assume that the car is behind door no.
  1, the host has a choice if and only if the
  candidate selects the first door - because
  otherwise there is only one "goat door" left!
• We assume that if the host has a choice, he will
  randomly select the door to open.  
• Candidate chooses Host opens p  
• Door 1 Door 2 1/6
• Door 1 Door 3 1/6
• Door 2 Door 3 1/3
• Door 3 Door 2 1/3
• Sum 1 

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Illustration-3

• candidate who always sticks to his original
  choice no matter what happens  
• Candidate chooses Host opens final choice win
  p  
• Door 1 Door 2 Door 1
  yes 1/6
• Door 1 Door 3 Door 1
  yes 1/6
• Door 2 Door 3 Door 2 no
  1/3
• Door 3 Door 2 Door 3 no
  1/3
• Sum1 Sum of cases where candidate wins 1/3 
• candidate who always switches to the other door
  whenever he gets the chance
•  Candidate chooses Host opens final choice win
  p  
• Door 1 Door 2 Door 3 no 1/6
• Door 1 Door 3 Door 2 no 1/6
• Door 2 Door 3 Door 1 yes 1/3
• Door 3 Door 2 Door 1 yes 1/3
• Sum1 Sum of cases where candidate wins
  2/3

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Key Issues

• The existing languages of probability do not
  really give us the syntax to express certain
  knowledge about the problem
• Lot of reasoning is done by the human being
• Our goal Develop a knowledge representation
  language and a reasoning system such that once we
  express our knowledge in that language the system
  can do the desired reasoning
• P-log is such an attempt

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Representing the Monty Hall problem in P-log.

• doors 1, 2, 3.
• open, selected, prize, D doors
• can_open(D) ?selected D.
• can_open(D) ? prize D.
• can_open(D) ?not can_open(D).
• pr(openD c can_open(D), can_open(D1), D / D1
  ) ½
• By default pr(open D c can_open(D) ) 1 when
  there is no D1, such that
• 
  
  can_open(D1) and D / D1 .
• random(prize), random(selected).
• random(open X can_open(X)).
• pr(prize D) 1/3.
• pr(selected D) 1/3.
• obs(selected 1). obs(open 2). obs(prize
  2).