Uncertainty in AI, Probabilistic reasoning(1) Especially for Bayesian Networks

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Uncertainty in AI, Probabilistic reasoning(1)
Especially for Bayesian Networks

• KyuTae Cho ,Jeong Ki Yoo ,HeeJin Lee

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Contents

• Uncertainty
• Degree of Belief / Degree of Truth
• Handling Uncertain Knowledge Probabilistic
  Reasoning System
• Basic Probability Theories Bayes rule
• Bayesian Network
• Concepts of Bayesian Networks
• Structure of Bayesian Networks
• Features of Bayesian Networks
• Evaluating Networks
• Initial probability of Bayesian Networks
• Conclusion

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Uncertainty

• Agents almost never have access to the whole
  truth about their environment
• Characteristics of real-world applications
• Truth value is unknown
• Too complex to compute prior to make decision
• Rational decision the right thing to do
• Depends on both the relative importance of
  various goals and the likelihood that, and degree
  to which, they will be achieved

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Degree of Belief

• Agent can provide degree of belief for sentence
• Main tool Probability theory
• assign a numerical degree of belief between 0 and
  1 to sentences
• the way of summarizing the uncertainty that comes
  from laziness and ignorance
• Probability can be derived from statistical data

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Degree of Belief vs. Degree of Truth

• Degree of Belief
• The sentence itself is in fact either true or
  false
• Same ontological commitment as logic the facts
  either do or do not hold in the world
• Probability theory
• Degree of Truth (membership)
• Not a question of the external world
• Case of vagueness or uncertainty about the
  meaning of the linguistic term tall, pretty
• Fuzzy set theory, fuzzy logic

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Handling Uncertain Knowledge

• Diagnosis Rule
• ?p Symptom(p, Toothache) ?Disease(p, Cavity)
• ?p Symptom(p, Toothache) ?Disease(p, Cavity) ?
  Disease(p,GumDisease) ?
  Disease(p, ImpactedWisdom)?
• Pr( Symptom Disease)
• Causal Rule
• ?p Disease(p, Cavity) ? Symptom(p, Toothache)
• Not every Cavity causes toothache
• Pr (Disease Symptom)

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Why First-order Logic Fails?

• Laziness Too much works to prepare complete set
  of exceptionless rule, and too hard to use the
  enormous rules
• Theoretical ignorance Medical science has no
  complete theory for the domain
• Practical ignorance All the necessary tests
  cannot be run, even though we know all the rules

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Probabilistic Reasoning System

• Assign probability to a proposition based on the
  percepts that it has received to date
• Evidence perception that an agent receives
• Probabilities can change when more evidence is
  acquired
• Prior / unconditional probability no evidence
  at all
• Posterior / conditional probability after
  evidence is obtained

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Uncertainty and Rational Decisions

• No plan can guarantee to achieve the goal
• To make choice, agent must have preferences
  between the different possible outcomes of
  various plans
• missing plane v.s. long waiting
• Utility theory to represent and reason with
  preferences
• Utility the quality of being useful (degree of
  usefulness)
• Decision Theory Probability Theory Utility
  Theory
• Principle of Maximum Expected Utility An agent
  is rational if and only if it chooses the action
  that yields the highest expected utility, average
  over all possible outcomes of the action

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Basic Probability

• Prior probability P(A) unconditional or prior
  probability that the proposition A is true
• Conditional Probability
• P(ab) P(a,b)/P(b)
• Product rule -gt P(a,b)P(b)
• The axioms of Probability (Kolmogorovs axioms)
• 0lt P(a) lt 1, for any proposition a
• P(true) 1 , P(false) 0
• P(a?b) p(a)p(b) p(a,b)

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Basic Probability

• The probability of a proposition is equal to the
  sum of the probabilities of the atomic events in
  which it holds
• P(a)
• where e(a) is set of all the atomic events in
  which a holds
• Marginalization, summing out
• P(Y) P(Y,z)
• Conditioning
• P(Y) P(Yz)P(z)
• independence between a and b
• P(ab) P(a)
• P(ba) P(b)
• P(a,b) P(a)P(b)

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Bayes rule

• P(ba) P(ab)P(b)/P(a)
• P(ba,e) P(ab,e)P(be)/P(ae) where e is the
  background evidence
• s patients having stiff neck
• m patients having meningtis
• P(sm) 0.5, P(m) 1/50000, P(s) 1/20
• P(ms) P(sm)P(m)/P(s) 0.5 x 1/50000 / 1/20
  0.0002

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Conditional Independence

• Conditional independence of two variables X and
  Y, given a third variable Z
• P(X,YZ) P(XZ)P(YZ)
• X,Y,Z are related with each other, but once the
  value of Z is settled, X and Y become independent
  between them.
• P(Cavitytoothache,catch) aP(toothache,catchcav
  ity)P(cavity)
• a P(toothachecavity) P(catchcavity)
  P(cavity)
• toothache and catch are directly caused by the
  cavity, but neither has a direct effect on the
  other.

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naive Bayes model

• a single cause directly influences a number of
  effects, all of which are conditionally
  independent, given cause.
• P(Cause,Effect1,..., EffectN)
  P(Cause)P(Effect1Cause)... P(EffectNCause)
• Naive because it is often used in cases where the
  effect variables are not conditionally
  independent given cause variable.
• Though work surprisingly well in practice

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• Bayesian Networks

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Bayesian Networks

• Concepts of Bayesian Networks
• Structure of Bayesian Networks
• Features of Bayesian Networks
• Evaluating Networks
• Initial probability of Bayesian Networks

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Concepts

• Model for representing uncertainty in our
  knowledge
• Graphical model of causality and influence
• Representation of the dependencies among random
  variables

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• Bayesian Networks
• Concepts of Bayesian Networks
• Structure of Bayesian Networks
• Features of Bayesian Networks
• Evaluating Networks
• Initial probability of Bayesian Networks

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Structure

• Form
• DAGs with following properties
• Nodes are random variables.
• Certain independence assumptions hold
• Components
• Nodes
• Random variables
• Arcs
• Specification of dependency among random
  variables.
• Probability distribution
• P(a) or P(ab)

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Structure(cont.)

• Initial configuration of BN
• Root nodes
• Prior probabilities
• Nonroot nodes
• Conditional probabilities given all possible
  combinations of direct predecessors

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Structure Example
P(fo) .15
P(bp) .01
Family-out(fo)
Bowel-problem(bp)
P(do fo bp) .99 P(do fo?bp).90 P(do ?fo
bp) .97 P(do?fo?bp).3
Dog-out(do)
Light-on(lo)
P(lofo) .6 P(lo?fo).05
Hear-bark(hb)
P(hbdo) .7 P(hb?do).05
lt Family-out problem gt
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• Bayesian Networks
• Concepts of Bayesian Networks
• Structure of Bayesian Networks
• Features of Bayesian Networks
• Indepenence Assumptions
• Consistent Probabilities
• Evaluating Networks
• Initial probability of Bayesian Networks

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Features

• Independence assumptions
• Relating to casual interpretation of arcs
• consistent probabilities
• Relating to the probabilities that are specified

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Independence assumptions

• Problem of probability theory
• 2n-1 joint distributions for n variables
• For 5 variables, 31 joint distributions
• Solution by BN
• For 5 variables, 10 joint distributions
• Bayesian Networks have built-in independence
  assumptions.

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Independence assumption(cont.)

• Definition of independence assumptions
• If random variable a,b are independent,
  P(ab) P(a)
• How to know dependency between two variables
• By the existence of d-connecting path between two
  random variables
• d-connecting path exist dependent

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Independence assumption D-connecting path
1) It is linear or diverging and not a member
of evidence nodes 2) It is converging, and
either n or one of its descendents is a
member of evidence nodes
A
X
.
.
intermediate nodes
.
Y
B
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Independence assumption(cont.)

• Example

A is dependent on B
B
A
B
A
Evidence node
C
D
D
E
Evidence node
C is independent on G
Evidence node
G
F
F
H
I
H
I
H is independent on I
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Consistent Probabilities

• Problem of probability theory
• Inconsistent probability
• Exgt P(ab).7 ,P(ba).3, P(b).5
• P(a) P(b)P(ab) / p(ba) .5
  .7 / .3 .35 / .3
• Solution by BN
• Bayesian Network has consistent probabilities
• Consistent numbers
• Unique definition of distribution

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Consistent Probabilities Joint
distribution

• Definition of joint distribution
• Set of boolean variables (a,b)
  (a,b) P(ab) P(?ab) P(a?b) P(?a?b)
• Role of joint distribution
• Joint distribution give all the information about
  probability distribution.
• Exgt P(ab) P(ab) / P(b)
• P(ab) / ((P(ab)P(?ab))
• For n random variables, 2n 1 joint
  distributions

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Consistent Probabilities Unique definition of
distribution

• Joint distribution for BN is uniquely defined
• By the product individual distribution of R.V.
• Using chain-rule, topological sort and dependency

Ex)
P(abcde) P(a)P(b)P(ca)P(dab)P(ed)
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Consistent Probabilities Unique definition of
distribution(cont.)

• Chain-rule
• Topological sort
• Ordering of variable comes before all its
  descendants.
• Exgt

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Consistent Probabilities Unique definition of
distribution(cont.)

• Dependency
• If no given nodes, node is only dependent on
  immediately above the node

Ex)
b is independent on a,c d is independent on c e
is independent on a,b,c
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Consistent Probabilities Unique definition of
distribution(cont.)

• Example

Chain-rule, Topological sort
Joint probability P(abcde)
P(a)P(ba)P(cab)P(dabc)P(eabcd)
Independence assumption
P(abcde) P(a)P(b)P(ca)P(dab)P(ed)
b is independent on a,c d is independent on c e
is independent on a,b,c
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• Bayesian Networks
• Concepts of Bayesian Networks
• Structure of Bayesian Networks
• Features of Bayesian Networks
• Evaluating Networks
• Exact Inference
• Approximate Solutions
• Initial probability of Bayesian Networks

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Evaluating networks

• Evaluation of network
• Computation of all nodes conditional probability
  given evidence
• Type of evaluation
• Exact inference
• NP-Hard Problem
• Approximate inference
• Not exact, but within small distance of the
  correct answer

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Exact inference

• Two network types
• Singly connected network (polytree)
• Multiply connected network
• Complexity according to network type
• Singly connected network can be efficiently solved

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Exact inference (cont.)

• Inference by enumeration (with alarm example)

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Exact inference (cont.)

• Variable elimination
• IDEA Do the calculation once and reuse later
• factor

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Exact inference Multiply Connected Network

• Hard to evaluate multiply connection network

A
Probabilities can be affected by both neighbor
nodes and other nodes
p(CD) ?
B
C
D
evidence
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Exact inference Multiply Connected Network
(cont.)

• Methodology to evaluate the network exactly
• Clustering
• To Combination of nodes until the resulting
  graph is singly connected

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Approximate Solutions

• Logic Sampling
• While(not assign to all values)
• guess the value of next lower node
• on the basis of the higher node values
• ex)

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• Bayesian Networks
• Concepts of Bayesian Networks
• Structure of Bayesian Networks
• Features of Bayesian Networks
• Evaluating Networks
• Initial probability of Bayesian Networks

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Initial probability of B.N.

• Determined by expert who subjectively assesses
  problems
• Ex) Three causes of a fever

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Conclusion

• Bayesian Networks are solutions of problems in
  traditional probability theory
• Drawback and choice
• Evaluation time is drawback of BN.
• Reason of using BN
• BN need not many numbers
• Efficient exact solution methods as well as a
  variety of approximation schemes