§7 Modelling Uncertainty Bayes’ theorem §7 Modelling Uncertainty Bayes’ theorem � probabilistic uncertainty probabilistic uncertainty � hypothesis hypothesis H H � � � probability of an outcome probability of an outcome � � evidence evidence E E � � dice, shuffled cards dice, shuffled cards � � probability of the hypothesis probability of the hypothesis P P ( ( H H ) ) � � statistical reasoning statistical reasoning � � probability of the evidence probability of the evidence P P ( ( E E ) ) � Bayesian networks, Dempster Bayesian networks, Dempster- -Shafer theory Shafer theory � � � possibilistic uncertainty possibilistic uncertainty � probability of the hypothesis based on the probability of the hypothesis based on the � � � possibility of classifying object possibility of classifying object evidence evidence � � sorites sorites paradoxes paradoxes P ( P ( H H | | E E ) = ( ) = ( P P ( ( E E | | H H ) ) · · P P ( ( H H )) / )) / P P ( ( E E ) ) � � fuzzy sets fuzzy sets � Example Example (cont’d) Example Example (cont’d) � P P ( ( H H ) = 0.10 ) = 0.10 � H H — — there is a bug in the code there is a bug in the code � � � P P ( ( E E | | H H ) = 0.90 ) = 0.90 � E E — — a bug is detected in the test a bug is detected in the test � � � P P ( ( E E | |¬ ¬ H H ) = 0.10 ) = 0.10 � � E E | | H H — — a bug is detected in the test given that a bug is detected in the test given that � � P P ( ( E E ) = ) = P P ( ( E E | | H H ) ) · · P P ( ( H H ) + ) + P P ( ( E E |¬ |¬ H H ) · ) · P P (¬ (¬ H H ) ) there is a bug in the code there is a bug in the code � = 0.18 = 0.18 � H H | | E E — — there is a bug in the code given that a there is a bug in the code given that a � � from Bayes’ theorem: from Bayes’ theorem: � bug is detected in the test bug is detected in the test 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 Dempster- Dempster -Shafer theory Shafer theory � describe cause describe cause- -and and- -effect relationships with a effect relationships with a � belief about a proposition as an interval belief about a proposition as an interval � � directed graph directed graph [ belief, plausability ] ⊆ ⊆ [ 0, 1] [ 0, 1] [ belief, plausability ] � vertices = propositions or variables vertices = propositions or variables � � belief supporting belief supporting A A : Bel( : Bel( A A ) ) � � edges = dependencies as probabilities edges = dependencies as probabilities � � plausability of plausability of A A : Pl( : Pl( A A ) = 1 ) = 1 − − Bel( Bel(¬ ¬ A A ) ) � � propagation of the probabilities propagation of the probabilities � � Bel(intruder) = 0.3, Pl(intruder) = 0.8 Bel(intruder) = 0.3, Pl(intruder) = 0.8 � � problems: problems: � � Bel(no intruder) = 0.2 Bel(no intruder) = 0.2 � � relationships between the evidence and hypotheses relationships between the evidence and hypotheses � are known are known � 0.5 of the probability range 0.5 of the probability range � is indeterminate is indeterminate � establishing and updating the probabilities establishing and updating the probabilities � 1
Belief interval Fuzzy sets Belief interval 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 µ µ A ( x x ) ) defined by a membership function A ( Belief Uncertainty Non-belief � not in the set: not in the set: µ µ A ( x x ) = 0 ) = 0 A ( � � fully in the set: fully in the set: µ µ A A ( ( x x ) = 1 ) = 1 � Plausability � partially in the set: 0 < partially in the set: 0 < µ µ A A ( ( x x ) < 1 ) < 1 � Doubt 0 0 Bel( Bel( A A ) ) Pl( A Pl( A ) ) 1 1 Membership function Fuzzy operations Membership function Fuzzy operations µ µ � union: union: µ µ C C ( ( x x ) = max{ ) = max{µ µ A A ( ( x x ), ), µ µ B B ( ( x x )} )} � 1 1 � intersection: intersection: µ µ C C ( ( x x ) = min{ ) = min{µ µ A A ( ( x x ), ), µ µ B B ( ( x x )} )} � A A � complement: complement: µ µ C C ( ( x x ) = 1 − ) = 1 − µ µ A A ( ( x x ) ) µ A ( x x ) ) µ A ( � � note: operations can be defined differently note: operations can be defined differently � 0 0 x U U Fuzzy operations (cont’d) Fuzzy operations (cont’d) Uses for fuzzy sets Uses for fuzzy sets µ µ � approximate reasoning approximate reasoning A ∪ ∪ B A B � 1 1 � fuzzy constraint satisfaction problem fuzzy constraint satisfaction problem � A A B B � fuzzy numbers fuzzy numbers A � � almost any ‘crisp’ method can be fuzzified! almost any ‘crisp’ method can be fuzzified! � A ∩ ∩ B A B 0 0 U U 2
Outroduction The intention, huh? Outroduction The intention, huh? §1 §1 Introduction Introduction §2 §2 Random Numbers Random Numbers � 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 §3 §3 Tournaments Tournaments computer scientist computer scientist §4 §4 Game Trees Game Trees §5 Path Finding §5 Path Finding §6 Decision- -Making Making §6 Decision §7 §7 Modelling Uncertainty Modelling Uncertainty Examinations Examination questions Examinations Examination questions examination dates (to be confirmed) examination dates (to be confirmed) questions questions � � � � October 26, 2005 October 26, 2005 1. 1. � based on both lectures and lecture notes based on both lectures and lecture notes � N.B. lecture examination, 12:00 − 14:00 lecture examination, 12:00 − 14:00 N.B. – – � two questions, à 5 points two questions, à 5 points � November 21, 2005 November 21, 2005 2. 2. � to pass the examination, at least 5 points (50%) are to pass the examination, at least 5 points (50%) are � January 30, 2006 January 30, 2006 3. 3. required required check the exact times and places at check the exact times and places at � � ⎡ p ⎤ = ⎡ 5 ⎤ � grade: grade: g g = p − − 5 � http://www.it.utu.fi/opetus/tentit/ http://www.it.utu.fi/opetus/tentit/ � questions are in English, but you can answer in questions are in English, but you can answer in � remember to enrol! remember to enrol! � � English or in Finnish English or in Finnish https://www.it.utu.fi/kurssi- https://www.it.utu.fi/kurssi -ilmo/ ilmo/ My two cents My two cents 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? off off- -the the- -shelf components: gfx cards, 3d engines, shelf components: gfx cards, 3d engines, animation tools, audio, AI, networking… animation tools, audio, AI, networking… mobile platforms: location- -based games based games mobile platforms: location 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 independent game publishing: war against apathy! independent game publishing: war against apathy! technology breeds new ideas— technology breeds new ideas —or does it? or does it? 3
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