Decision Making Under Uncertainty

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Decision Making Under Uncertainty

• CSE 495
• Resources
• Russell and Norwicks book

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Some Examples
1. Suppose that you are in a TV show and you have
already earned 1000.000 so far. Now, the
presentator propose you a gamble he will flip a
coin if the coin comes up heads you will earn
3000.000. But if it comes up tails you will
loose the 1000.000. What do you decide?

• 2. Considerations for sitting a new airport
• Cost
• noise pollution
• Safety
• If there are two candidate sites, how to decide
  between them?

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First Order Representation
?p symptom(p,toothache) ? disease(p, cavity)
?p disease(p, cavity) ? symptom(p,toothache)
For the dentist in you, which one is true?
None. A better rule could be something like
?p symptom(p,toothache) ?? disease(p, cavity) ?

disease(p, Gumdisease) ?
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Problems with First Order Representation

• Laziness it is too much work to list antecedents
  and consequents
• Theoretical Ignorance not all known
  antecedents/consequents may be known
• Practical ignorance even if we know all the
  rules, uncertainty in establishing
  antecedents/consequents may still occur

Related to the knowledge-acquisition bottle neck
Fuzzy Logic
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Handling Uncertain Knowledge

• We have only a degree of belief
• Probability theory can be used to deal with
  degree of belief
• If the probability of an event is 1 does it mean
  that the event will occur?

No. It means that we believe that will always
occur

• In our example we may believe that with 0.8
  probability the patient has cavities

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Uncertainty and Rational Decisions

• In addition to probabilities there might be
  preferences

Plan 1 Take the train from NYC to NC at 700AM
(probability of missing a connection in WAS 5,
waiting time in WAS 4 hours) Plan 2 Take the
train from NYC to NC at 1000AM (probability of
missing a connection in WAS 35, waiting time
in WAS 1 hour)
Which would you choose?
The point being that decisions are not made based
only on the probability of events
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Decision Theory

• Utility theory represents and reasons with
  preferences

Decision Theory probability theory utility
theory
1. Calculate probabilities of current state 2.
Calculate probabilities of action 3. Select
actions with the highest expected utility
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Probability Distribution

• The events E1, E2, , Ek must meet the following
  conditions
• One always occur
• No two can occur at the same time
• The probabilities p1, , pn are numbers
  associated with these events, such that 0 ? pi ?
  1 and p1 pn 1
• A probability distribution assigns probabilities
  to events such that the two properties above holds

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Expected Value
In general, let Q be a quantity that has value v1
with probability p1, ., vk with probability pk
then the Expected value of Q is
p1 v1 p2 v2 pk vK
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Selection of a Good Attribute Information Gain
Theory

• If the possible answers vi have probabilities
  p(vi), then the information content of the actual
  answer is given by

• Examples
• Information content with the fair coin
• Information content with the totally unfair
• Information content with the very unfair

I(1/2,1/2) 1
I(1,0) 0
I(1/100,99/100) 0.08
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Uniform Distribution
A probability distribution is uniform if there
are k events each of which has probability 1/k
Examples?
Rolling a fair dice. The events being that the
dice will comes up on each of the dices faces
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Probability of Two Events Taken Place
Two events are independent if the occurrence of
one doesnt affect the occurrence of the other
one.
If E1 and E2 are two independent events, then the
probability that E1 and E2 occur is p(E1)p(E2)
If the probability of having a winning a lottery
is .1, then the probability of winning the
lottery two times in a row is .1.1 .01
If E1 and E2 are two independent events, then the
probability that E1 or E2 occur is p(E1) p(E2)
(p(E1)p(E2))
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Conditional Probability
P(A ? B) P(B)P(AB) P(A)P(BA)
(product rule)
P(AB) (P(A)P(BA))/ P(B)
(Bayes rule)

• Example Suppose that the following is true
• Meningitis cause stiff neck, 50 of the time
• Probability of patient having meningitis is
  1/50000
• Probability of patient having stiff neck is 1/120

P(MS)
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Axioms of Probability

1. For any event A, 0 ? P(A) ? 1 holds
2. If F can never occur then P(F) 0 and if T
   always occurs then P(T) 1
3. P(E1? E2) p(E1) p(E2) p(E1?E2)

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Axioms of Probability (2)

• Suppose there is a betting game between 2 persons
  for money.
• Suppose that person 1 has some degree of belief
  in an event A.
• I bet 6 that it will occur. My degree of
  belief is 0.4
• Suppose that person 2 bets for or against A
  consistent with the degree of belief of person 1.

I bet 4 that it will not occur
Theorem (Bruno de Finetti, 1931) If Person 1
sets of degrees violating the axioms of the
probability, then there is a betting strategy for
Person 2 that guarantees that Person 1 looses
money