7 Modelling Uncertainty Bayes theorem 7 Modelling Uncertainty - - PDF document

SLIDE 1

1

§7 Modelling Uncertainty §7 Modelling Uncertainty

• probabilistic uncertainty

probabilistic uncertainty

• probability of an outcome

probability of an outcome

• dice, shuffled cards

dice, shuffled cards

• statistical reasoning

statistical reasoning

• Bayesian networks, Dempster

Bayesian networks, Dempster-

• Shafer theory

Shafer theory

• possibilistic uncertainty

possibilistic uncertainty

• possibility of classifying object

possibility of classifying object

• sorites

sorites paradoxes paradoxes

• fuzzy sets

fuzzy sets

Bayes’ theorem Bayes’ theorem

• hypothesis

hypothesis H H

• evidence

evidence E E

• probability of the hypothesis

probability of the hypothesis P P( (H H) )

• probability of the evidence

probability of the evidence P P( (E E) )

• probability of the hypothesis based on the

probability of the hypothesis based on the evidence evidence P P( (H H| |E E) = ( ) = (P P( (E E| |H H) ) · · P P( (H H)) / )) / P P( (E E) )

Example Example

• H

H — — there is a bug in the code there is a bug in the code

• E

E — — a bug is detected in the test a bug is detected in the test

• E

E| |H H — — a bug is detected in the test given that a bug is detected in the test given that there is a bug in the code there is a bug in the code

• H

H| |E E — — there is a bug in the code given that a there is a bug in the code given that a bug is detected in the test bug is detected in the test

Example (cont’d) Example (cont’d)

• P

P( (H H) = 0.10 ) = 0.10

• P

P( (E E| |H H) = 0.90 ) = 0.90

• P

P( (E E| |¬ ¬H H) = 0.10 ) = 0.10

• P

P( (E E) = ) = P P( (E E| |H H) ) · · P P( (H H) + ) + P P( (E E|¬ |¬H H) · ) · P P(¬ (¬H H) ) = 0.18 = 0.18

• from Bayes’ theorem:

from Bayes’ theorem: P P( (H H| |E E) = 0.5 ) = 0.5

• conclusion: a detected bug has fifty

conclusion: a detected bug has fifty-

• fifty chance

fifty chance that it is not in the actual code that it is not in the actual code

Bayesian networks Bayesian networks

• describe cause

describe cause-

• and

and-

• effect relationships with a

effect relationships with a directed graph directed graph

• vertices = propositions or variables

vertices = propositions or variables

• edges = dependencies as probabilities

edges = dependencies as probabilities

• propagation of the probabilities

propagation of the probabilities

• problems:

problems:

• relationships between the evidence and hypotheses

relationships between the evidence and hypotheses are known are known

• establishing and updating the probabilities

establishing and updating the probabilities

Dempster Dempster-

• Shafer theory

Shafer theory

• belief about a proposition as an interval

belief about a proposition as an interval [ belief, plausability ] [ belief, plausability ] ⊆ ⊆ [ 0, 1] [ 0, 1]

• belief supporting

belief supporting A A: Bel( : Bel(A A) )

• plausability of

plausability of A A: Pl( : Pl(A A) = 1 ) = 1 − − Bel( Bel(¬ ¬A A) )

• Bel(intruder) = 0.3, Pl(intruder) = 0.8

Bel(intruder) = 0.3, Pl(intruder) = 0.8

• Bel(no intruder) = 0.2

Bel(no intruder) = 0.2

• 0.5 of the probability range

0.5 of the probability range is indeterminate is indeterminate

SLIDE 2

2

Belief interval Belief interval

1 1 Bel( Bel(A A) ) Pl( Pl(A A) ) Belief Uncertainty Non-belief Plausability Doubt

Fuzzy sets Fuzzy sets

• element

element x x has a membership in the set has a membership in the set A A defined by a membership function defined by a membership function µ µA

A(

(x x) )

• not in the set:

not in the set: µ µA

A(

(x x) = 0 ) = 0

• fully in the set:

fully in the set: µ µA

A(

(x x) = 1 ) = 1

• partially in the set: 0 <

partially in the set: 0 < µ µA

A(

(x x) < 1 ) < 1

Membership function Membership function

U U µ µ 1 1 A A x µ µA

A(

(x x) )

Fuzzy operations Fuzzy operations

• union:

union: µ µC

C(

(x x) = max{ ) = max{µ µA

A(

(x x), ), µ µB

B(

(x x)} )}

• intersection:

intersection: µ µC

C(

(x x) = min{ ) = min{µ µA

A(

(x x), ), µ µB

B(

(x x)} )}

• complement:

complement: µ µC

C(

(x x) = 1 − ) = 1 − µ µA

A(

(x x) )

• note: operations can be defined differently

note: operations can be defined differently

Fuzzy operations (cont’d) Fuzzy operations (cont’d)

U U µ µ 1 1 A A B B A A ∪ ∪ B B A A ∩ ∩ B B A

Uses for fuzzy sets Uses for fuzzy sets

• approximate reasoning

approximate reasoning

• fuzzy constraint satisfaction problem

fuzzy constraint satisfaction problem

• fuzzy numbers

fuzzy numbers

• almost any ‘crisp’ method can be fuzzified!

almost any ‘crisp’ method can be fuzzified!

SLIDE 3

3

Outroduction Outroduction

§1 §1 Introduction Introduction §2 §2 Random Numbers Random Numbers §3 §3 Tournaments Tournaments §4 §4 Game Trees Game Trees §5 §5 Path Finding Path Finding §6 §6 Decision Decision-

• Making

Making §7 §7 Modelling Uncertainty Modelling Uncertainty

The intention, huh? The intention, huh?

• to provide a glance into the world of computer

to provide a glance into the world of computer games as seen from the perspective of a games as seen from the perspective of a computer scientist computer scientist

Examinations Examinations

• examination dates (to be confirmed)

examination dates (to be confirmed)

1. 1.

October 26, 2005 October 26, 2005

– –

N.B. N.B. lecture examination, 12:00−14:00

lecture examination, 12:00−14:00

2. 2.

November 21, 2005 November 21, 2005

3. 3.

January 30, 2006 January 30, 2006

• check the exact times and places at

check the exact times and places at http://www.it.utu.fi/opetus/tentit/ http://www.it.utu.fi/opetus/tentit/

• remember to enrol!

remember to enrol! https://www.it.utu.fi/kurssi https://www.it.utu.fi/kurssi-

• ilmo/

ilmo/

Examination questions Examination questions

• questions

questions

• based on both lectures and lecture notes

based on both lectures and lecture notes

• two questions, à 5 points

two questions, à 5 points

• to pass the examination, at least 5 points (50%) are

to pass the examination, at least 5 points (50%) are required required

• grade:

grade: g g = = ⎡ ⎡p p − − 5 5⎤ ⎤

• questions are in English, but you can answer in

questions are in English, but you can answer in English or in Finnish English or in Finnish

My two cents My two cents

independent game publishing: war against apathy! independent game publishing: war against apathy! mobile platforms: location mobile platforms: location-

• based games

based games software construction practices: will game programming software construction practices: will game programming remain the last reservate for wizards, nerds and geeks? remain the last reservate for wizards, nerds and geeks?

• ff
• ff-
• the

the-

• shelf components: gfx cards, 3d engines,

shelf components: gfx cards, 3d engines, animation tools, audio, AI, networking… animation tools, audio, AI, networking… technology breeds new ideas technology breeds new ideas— —or does it?

• r does it?

untapped markets: not every game buyer is (nor even untapped markets: not every game buyer is (nor even don’t want to be) familiar with current game genres don’t want to be) familiar with current game genres