Uncertainty in AI

1
Uncertainty in AI

• Outline
• Introduction
• Basic Probability Theory
• Probabilistic Reasoning
• Why should we use probability theory?
• Dutch Book Theorem

2
Sources of Uncertainty

• Information is partial
• Information is not fully reliable.
• Representation language is inherently imprecise.
• Information comes from multiple sources and it is
  conflicting.
• Information is approximate
• Non-absolute cause-effect relationships exist

3
Basic Probability

• Probability theory enables us to make rational
  decisions.
• Which mode of transportation is safer
• Car or Plane?
• What is the probability of an accident?

4
Basic Probability Theory

• An experiment has a set of potential outcomes,
  e.g., throw a dice
• The sample space of an experiment is the set of
  all possible outcomes, e.g., 1, 2, 3, 4, 5, 6
• An event is a subset of the sample space.
• 2
• 3, 6
• even 2, 4, 6
• odd 1, 3, 5

5
Probability as Relative Frequency

• An event has a probability.
• Consider a long sequence of experiments. If we
  look at the number of times a particular event
  occurs in that sequence, and compare it to the
  total number of experiments, we can compute a
  ratio.
• This ratio is one way of estimating the
  probability of the event.
• P(E) ( of times E occurred)/(total of trials)

6

• Example
• 100 attempts are made to swim a length in 30
  secs. The swimmer succeeds on 20 occasions
  therefore the probability that a swimmer can
  complete the length in 30 secs is
• 20/100 0.2
• Failure 1-.2 or 0.8
• The experiments, the sample space and the events
  must be defined clearly for probability to be
  meaningful
• What is the probability of an accident?

7
Theoretical Probability

• Principle of IndifferenceAlternatives are always
  to be judged equiprobable if we have no reason to
  expect or prefer one over the other.
• Each outcome in the sample space is assigned
  equal probability.
• Example throw a dice
• P(1)P(2) ... P(6)1/6

8
Law of Large Numbers

• As the number of experiments increases the
  relative frequency of an event more closely
  approximates the theoretical probability of the
  event.
• if the theoretical assumptions hold.
• Buffons Needle for Computing p
• Draw parallel lines 1 inch apart on a plane
• Throw a 1-inch needle on the plane
• P( needle crossing a line )2/p

9
Large Number Reveals Untruth in Assumptions

• Results of 1,000,000 throws of a die
• Number 1 2 3 4 5 6
• Fraction .155 .159 .164 .169 .174 .179

10
Axioms of Probability Theory

• Suppose P(.) is a probability function, then
• 1. for any event E, 0P(E) 1.
• 2. P(S) 1, where S is the sample space.
• 3. for any two mutually exclusive events E1 and
  E2,
• P(E1 È E2) P(E1) P(E2)
• Any function that satisfies the above three
  axioms is a probability function.

11
Joint Probability

• Let A, B be two events, the joint probability of
  both A and B being true is denoted by P(A, B).
• Example
• P(spade) is the probability of the top card
  being a spade.
• P(king) is the probability of the top card being
  a king.
• P(spade, king) is the probability of the top
  card being both a spade and a king, i.e., the
  king of spade.
• P(king, spade)P(spade, king) ???

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Properties of Probability

• 1. P(ØE) 1 P(E)
• 2. If E1 and E2 are logically equivalent, then
• P(E1)P(E2).
• E1 Not all philosophers are more than six feet
  tall.
• E2 Some philosopher is not more that six feet
  tall.
• Then P(E1)P(E2).
• 3. P(E1, E2)P(E1).

13
Conditional Probability

• The probability of an event may change after
  knowing another event.
• The probability of A given B is denoted by
  P(AB).
• Example
• P( Wspace ) the probability of a randomly
  selected word from an English text is space
• P( Wspace Wouter) the probability of space
  if the previous word is outer

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Example

• A the top card of a deck of poker cards is a
  king of spade
• P(A) 1/52
• However, if we know
• B the top card is a king
• then, the probability of A given B is true is
• P(AB) 1/4.

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How to Compute P(AB)?
B
A
16
Business Students

• Of 100 students completing a course, 20 were
  business major. Ten students received As in the
  course, and three of these were business majors.,
  suppose A is the event that a randomly selected
  student got an A in the course, B is the event
  that a randomly selected event is a business
  major. What is the probability of A? What is the
  probability of A after knowing B is true?

17
Probabilistic Reasoning

• Evidence
• What we know about a situation.
• Hypothesis
• What we want to conclude.
• Compute
• P( Hypothesis Evidence )

18
Credit Card Authorization

• E is the data about the applicant's age, job,
  education, income, credit history, etc,
• H is the hypothesis that the credit card will
  provide positive return.
• The decision of whether to issue the credit card
  to the applicant is based on the probability
  P(HE).

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Medical Diagnosis

• E is a set of symptoms, such as, coughing,
  sneezing, headache, ...
• H is a disorder, e.g., common cold, SARS, flu.
• The diagnosis problem is to find an H (disorder)
  such that P(HE) is maximum.

20

• Linda is 31 years old, single, outspoken, and
  very bright. She majored in philosophy. As a
  student, she was deeply concerned with issues of
  discrimination and social justice, and also
  participated in antinuclear demonstrations.
• Please rank the following statements by their
  probability, using 1 for the most probable and 8
  for the least probable.
• a. Linda is a teacher in elementary school.
• b. Linda works in a bookstore and takes yoga
  classes.
• c. Linda is active in the feminist movement.
• d. Linda is psychiatric social worker.
• e. Linda is a member of the League of Women
  Voters.
• f. Linda is a bank teller.
• g. Linda is an insurance salesperson.
• h. Linda is a bank teller and is active in the
  feminist movement.

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Example
A patient takes a lab test and the result comes
back positive. The test has a false negative rate
of 2 and false positive rate of 3. Furthermore,
0.8 of the entire population have this
cancer. What is the probability of cancer if we
know the test result is positive?
22
Bayes Theorem

• If P(E2)gt0, then P(E1E2)P(E2E1)P(E1)/P(E2)
• This can be derived from the definition of
  conditional probability.

23
The Three-Card Problem

• Three cards are in a hat. One is red on both
  sides (the red-red card). One is white on both
  sides (the white-white card). One is red on one
  side and white on the other (the red-white card).
  A single card is drawn randomly and tossed into
  the air.
• a. What is the probability that the red-red card
  was drawn? (RR)
• b. What is the probability that the drawn cards
  lands with a white side up? (W-up)
• c. What is the probability that the red-red card
  was not drawn, assuming that the drawn card lands
  with the a red side up. (not-RRR-up)

24
Fair Bets

• A bet is fair to an individual I if, according to
  the individual's probability assessment, the bet
  will break even in the long run.
• The following three bet are fair
• Bet (a) Win 4.20 if RR
• lose 2.10
• otherwise. since you believe P(RR)1/3
• Bet (b) Win 2.00 if W-up
• lose 2.00
• otherwise. since you believe P(W-up)1/2
• Bet (c) Win 4.00 if R-up and not-RR
• lose 4.00 if R-up and RR
• neither win nor lose if not-R-up.
• since you believe P(not-RRR-up)1/2

25
Dutch Book

• The bets that you accepted have an interesting
  property
• No matter what card is drawn in the three-card
  problem, and no matter how it lands, you are
  guaranteed to lose money.
• This is called a Dutch Book

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Verification

• there are three possible outcomes
• 1. Some card other than red-red is drawn, and it
  lands with white side up. That is, W-up and
  not-RR
• 2. Some card other than red-red is drawn, and it
  lands with a red side up. That is, R-up and
  not-RR.
• 3. The red-red card is drawn, and it lands (of
  course) with a red side up. That is, R-up and RR.
• 1 2 3
• a. 2.10 2.10 4.20
• b. 2.00 2.00 2.00
• c. 0.00 4.00 4.00
• total 0.10 0.10 1.80

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The Dutch Book Theorem

• Suppose that an individual I is willing to accept
  any bet that is fair for I. Then a Dutch book can
  be made against I if and only if I's assessment
  of probability violates Bayesian axiomatization.

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Independence Intuition

• Events are independent if one has nothing
  whatever to do with others. Therefore, for two
  independent events, knowing one happening does
  change the probability of the other event
  happening.
• one toss of coin is independent of another coin
  (assuming it is a regular coin).
• price of tea in England is independent of the
  result of general election in Canada.

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Independent or Dependent?

• Getting cold and getting cat-allergy
• Mile Per Gallon and acceleration.
• Size of a persons vocabulary the persons shoe
  size.

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Independence Definition

• Events A and B are independent iff
• P(A, B) P(A) x P(B)
• which is equivalent to
• P(AB) P(A) and
• P(BA) P(B)
• when P(A, B) gt0.
• T1 the first toss is a head.
• T2 the second toss is a tail.
• P(T2T1) P(T2)

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Conditional Independence

• Dependent events can become independent given
  certain other events.
• Example,
• Size of shoe
• Age
• Size of vocabulary
• Two events A, B are conditionally independent
  given a third event C iff
• P(AB, C) P(AC)

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Conditional IndependenceDefinition

• Let E1 and E2 be two events, they are
  conditionally independent given E iff
• P(E1E, E2)P(E1E),
• that is the probability of E1 is not changed
  after knowing E2, given E is true.
• Equivalent formulations
• P(E1, E2E)P(E1E) P(E2E)
• P(E2E, E1)P(E2E)

33
Example Play Tennis?
Predict playing tennis when ltsunny, cool, high,
stronggt What probability should be used to make
the prediction? How to compute the probability?
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Probabilities of Individual Attributes

• Given the training set, we can compute the
  probabilities

35
Naïve Bayes Method

• Knowledge Base contains
• A set of hypotheses
• A set of evidences
• Probability of an evidence given a hypothesis
• Given
• A sub set of the evidences known to be present in
  a situation
• Find
• the hypothesis with the highest posterior
  probability P(HE1, E2, , Ek).
• The probability itself does not matter so much.

36
Naïve Bayes Method

• Assumptions
• Hypotheses are exhaustive and mutually exclusive
• H1 v H2 v v Hk
• (Hi Hj) for any i?j
• Evidences are conditionally independent given a
  hypothesis
• P(E1, E2,, EkH) P(E1H)P(EkH)
• P(H E1, E2,, Ek)
• P(E1, E2,, Ek, H)/P(E1, E2,, Ek)
• P(E1, E2,, EkH)P(H)/P(E1, E2,, Ek)

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Naïve Bayes Method

• The goal is to find H that maximize P(HE1, E2,,
  Ek)
• Since
• P(HE1, E2,, Ek) P(E1, E2,, EkH)P(H)/P(E1,
  E2,, Ek)
• and P(E1, E2,, Ek) is the same for different
  hypotheses,
• Maximizing P(HE1, E2,, Ek) is equivalent to
  maximizing P(E1, E2,, EkH)P(H)
  P(E1H)P(EkH)P(H)
• Naïve Bayes Method
• Find a hypothesis that maximizes
  P(E1H)P(EkH)P(H)

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Example Play Tennis

• P( sunny, cool, high, strong) vs.
• P(- sunny, cool, high, strong)
• P(sunny)P(cool)P(high)P(strong)P() vs.
• P(sunny-)P(cool-)P(high-)P(strong-)P(-)

39
Application Spam Detection

• Spam
• Dear sir, We want to transfer to overseas (
  126,000.000.00 USD) One hundred and Twenty six
  million United States Dollars) from a Bank in
  Africa, I want to ask you to quietly look for a
  reliable and honest person who will be capable
  and fit to provide either an existing
• Legitimate email
• Ham for lack of better name.

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• Hypotheses Spam, Ham
• Evidence a document
• The document is treated as a set (or bag) of
  words
• Knowledge
• P(Spam)
• The prior probability of an e-mail message being
  a spam.
• How to estimate this probability?
• P(wSpam)
• the probability that a word is w if we know w is
  chosen from a spam.
• How to estimate this probability?

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Limitations of Naïve Bayesian

• Cannot handle hypotheses of composite hypotheses
  well
• Suppose are independent of
  each other
• Consider a composite hypothesis
• How to compute the posterior probability

42

• Using the Bayes Theorem

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• but this is a very unreasonable assumption
• Need a better representation and a better
  assumption

E and B are independent But when A is given, they
are (adversely) dependent because they become
competitors to explain A P(BA, E) ltltP(BA) E
explains away of A
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• Cannot handle causal chaining
• Ex. A weather of the year
• B cotton production of the year
• C cotton price of next year
• Observed A influences C
• The influence is not direct (A -gt B -gt C)
• P(CB, A) P(CB) instantiation of B blocks
  influence of A on C

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Summary

• Basics of Probability Theory
• Experiment, sample space, events
• Axioms and prosperities
• Joint Probability
• Conditional Probability
• Probabilistic Reasoning
• Bayes Theorem
• Dutch Book Theorem
• Independence and Conditional Independence
• Naïve Bayes Method