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Probabilistic Reasoning; Network-based reasoning. COMPSCI 276 Fall 2007. Class Description. Instructor: Rina Dechter Days: Monday & Wednesday Time: 2:00 - 3:20 pm Class page: http://www.ics.uci.edu/~dechter/ics-275b/Fall-2007/. Why uncertainty. Summary of exceptions

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Probabilistic Reasoning; Network-based reasoning

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Probabilistic reasoning network based reasoning l.jpg

Probabilistic Reasoning;Network-based reasoning

COMPSCI 276

Fall 2007


Class description l.jpg

Class Description

  • Instructor: Rina Dechter

  • Days: Monday & Wednesday

  • Time: 2:00 - 3:20 pm

  • Class page:http://www.ics.uci.edu/~dechter/ics-275b/Fall-2007/


Why uncertainty l.jpg

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


The problem l.jpg

The problem

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(A|B) = 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

C

D

y

B

1.Parallel Combination:

CF(C) = x+y-xy, if x,y>0

CF(C) = (x+y)/(1-min(x,y)), x,y have different sign

CF( C) = x+y+xy, if x,y<0

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


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What’s in a rule?

A and BC

(m+n-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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P(S)

P(C|S)

P(B|S)

  • C B D=0 D=1

  • 0 0 0.1 0.9

  • 0 1 0.7 0.3

  • 1 0 0.8 0.2

  • 1 1 0.9 0.1

CPD:

P(X|C,S)

P(D|C,B)

Conditional Independencies

Efficient Representation

Bayesian Networks: Representation

Smoking

lung Cancer

Bronchitis

X-ray

Dyspnoea

P(S, C, B, X, D)

= P(S) P(C|S) P(B|S) P(X|C,S) P(D|C,B)


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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(A|K,B)=P(A|K)

  • 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


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