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Probabilistic Reasoning; Network-based reasoningPowerPoint Presentation

Probabilistic Reasoning; Network-based reasoning

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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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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
- 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:

AC

BC

---------

A and B - C

The problem

True

propositions

Uncertain

propositions

Q: Does T fly?

P(Q)?

Logic?....but how we handle exceptions

Probability: astronomical

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

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.

AB: m

P(A|B) = m

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

Burglery Example

Burglery

Phone

call

Alarm

Earthquake

Radio

AB

A more credible

------------------

B more credible

IF Alarm Burglery

A more credible (after radion)

But B is less credible

Rule from effect to causes

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.

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: RepresentationSmoking

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)

Markov and Bayesian Networks

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

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.

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

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.

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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