Probabilistic Reasoning; Network-based reasoning

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Probabilistic ReasoningNetwork-based reasoning

• COMPSCI 276
• Fall 2007

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Class Description

• Instructor Rina Dechter
• Days Monday Wednesday
• Time 200 - 320 pm
• Class page http//www.ics.uci.edu/dechter/ics-27
  5b/Fall-2007/

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Why uncertainty

• Summary of exceptions
• Birds fly, smoke means fire (cannot enumerate all
  exceptions.
• Why is it difficult?
• Exception combines in intricate ways
• e.g., we cannot tell from formulas how exceptions
  to rules interact

A?C B?C --------- A and B -? C
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The problem
All men are mortal T
All penguins are birds T

Socrates is a man
Men are kind p1
Birds fly p2
T looks like a penguin
Turn key gt car starts P_n
True propositions
Uncertain propositions
Q Does T fly? P(Q)?
Logic?....but how we handle exceptions Probability
astronomical
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Managing Uncertainty

• Knowledge obtained from people is almost always
  loaded with uncertainty
• Most rules have exceptions which one cannot
  afford to enumerate
• Antecedent conditions are ambiguously defined or
  hard to satisfy precisely
• First-generation expert systems combined
  uncertainties according to simple and uniform
  principle
• Lead to unpredictable and counterintuitive
  results
• Early days logicist, new-calculist,
  neo-probabilist

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Extensional vs Intensional Approaches

• Extensional (e.g., Mycin, Shortliffe, 1976)
  certainty factors attached to rules and combine
  in different ways.
• Intensional, semantic-based, probabilities are
  attached to set of worlds.

A?B m
P(AB) m
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Certainty combination in Mycin
A
x
If A then C (x) If B then C (y) If C then D (z)
z
D
C
y
B
1.Parallel Combination CF(C) xy-xy, if
x,ygt0 CF(C) (xy)/(1-min(x,y)), x,y have
different sign CF( C) xyxy, if x,ylt0 2.
Series combination 3.Conjunction, negation
Computational desire locality, detachment,
modularity
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Burglery Example
Burglery
Phone call
Alarm
Earthquake
Radio
A?B A more credible ------------------ B more
credible
IF Alarm ? Burglery A more credible (after
radion) But B is less credible
Rule from effect to causes
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Extensional vs Intensional
Extensional
Intensional
Uncertaintytruth value Uncertainty modality
Connectives combine certainty weight Connectives combine set of worlds
Rules Procedural license summary of a problem solving history Rules constraints on the world summary of world knowledge
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Whats in a rule?
A?B (m) C?B (n) P(BA) p A?B (p)
Semantic difficulties Handling exceptions, Retracting conclusions Unidirectional references Incoherent updating Semantic clarity Syntax mirrors world knowledge Empirically testable parameters Bidirectional Inferences Coherent updating
Computational merit Localitydetachment Computational difficulty Actions must weight verification of relevance
A and B?C (mn-mn)
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Why networks?

• Claim the basic steps invoked while people query
  and update their knowledge corresponds to mental
  tracings of pre-established links in dependency
  graphs
• Claim the degree to which an explanation mirrors
  these tracings determines whether it is
  psychologically meaningful.

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Bayesian Networks Representation
Smoking
lung Cancer
Bronchitis
X-ray
Dyspnoea
P(S) P(CS) P(BS) P(XC,S) P(DC,B)
P(S, C, B, X, D)
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Markov and Bayesian Networks

• Pearl Chapter 3
• (Read chapter 2 for background and refresher)

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The Qualitative Notion of Depedence

• The traditional definition of independence uses
  equality of numerical quantities as in
  P(x,y)P(x)P(y)
• People can easily and confidently detect
  dependencies, even though they may not be able to
  provide precise numerical estimates of
  probabilities.
• The notion of relevance and dependence are far
  more basic to human reasoning than the numerical
  values attached to probabilistic judgements.
• Should allow assertions about dependency
  relationships to be expressed qualitatively,
  directly and explicitly.
• Once asserted, these dependency relationships
  should remain a part of the representation
  scheme, impervious to variations in numerical
  inputs.

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The Qualitative Notion of Depedence(continue)

• Information about dependencies is essential in
  reasoning
• If we have acquired a body of knowledge K and now
  wish to assess the truth of proposition A, it is
  important to know whether it is worthwhile to
  consult another proposition B, which is not in K.
• How can we encode relevance information in a
  symbolic system?
• The number of (A,B,K) combinations is
  astronomical.
• Acquisition of new facts may destroy existing
  dependencies as well as create new ones
  (e.g.,age, hight,reading ability, or ground
  wet,rain,sprinkler)
• The first kind of change is called normal . The
  second will be called induced.
• Irrelevance is denoted P(AK,B)P(AK)
• Dependency relationships are qualitative and can
  be logical

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Dependency graphs

• The nodes represent propositional variables and
  the arcs represent local dependencies among
  conceptually related propositions.
• Explicitness, stability
• Graph concepts are entrenched in our language
  (e.g., thread of thoughts, lines of
  reasoning, connected ideas)
• One wonders if people can reason any other way
  except by tracing links and arrows and paths in
  some mental representation of concepts and
  relations.
• What types of dependencies and independencies are
  deducible from the topological properties of a
  graph?
• For a given probability distribution P and any
  three variables X,Y,Z,it is straightforward to
  verify whether knowing Z renders X independent of
  Y, but P does not dictates which variables should
  be regarded as neighbors.
• Some useful properties of dependencies and
  relevancies cannot be represented graphically.

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Why axiomatic characterization?

• Allow deriving conjectures about independencies
  that are clearer
• Axioms serve as inference rules
• Can capture the principal differences between
  various notions of relevance or independence