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chapter13 - Uncertainty Chapter 13 Chapter 13 1 Outline...

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Uncertainty Chapter 13 Chapter 13 1
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Outline Uncertainty Probability Syntax and Semantics Inference Independence and Bayes’ Rule Chapter 13 2
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Uncertainty Let action A t = leave for airport t minutes before flight Will A t get me there on time? Problems: 1) partial observability (road state, other drivers’ plans, etc.) 2) noisy sensors (KCBS tra ffi c reports) 3) uncertainty in action outcomes (flat tire, etc.) 4) immense complexity of modelling and predicting tra ffi c Hence a purely logical approach either 1) risks falsehood: “ A 25 will get me there on time” or 2) leads to conclusions that are too weak for decision making: A 25 will get me there on time if there’s no accident on the bridge and it doesn’t rain and my tires remain intact etc etc.” ( A 1440 might reasonably be said to get me there on time but I’d have to stay overnight in the airport . . . ) Chapter 13 3
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Methods for handling uncertainty Default or nonmonotonic logic: Assume my car does not have a flat tire Assume A 25 works unless contradicted by evidence Issues: What assumptions are reasonable? How to handle contradiction? Rules with fudge factors : A 25 0 . 3 AtAirportOnTime Sprinkler 0 . 99 WetGrass WetGrass 0 . 7 Rain Issues: Problems with combination, e.g., Sprinkler causes Rain ?? Probability Given the available evidence, A 25 will get me there on time with probability 0 . 04 Mahaviracarya (9th C.), Cardamo (1565) theory of gambling ( Fuzzy logic handles degree of truth NOT uncertainty e.g., WetGrass is true to degree 0 . 2 ) Chapter 13 4
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Probability Probabilistic assertions summarize e ff ects of laziness : failure to enumerate exceptions, qualifications, etc. ignorance : lack of relevant facts, initial conditions, etc. Subjective or Bayesian probability: Probabilities relate propositions to one’s own state of knowledge e.g., P ( A 25 | no reported accidents ) = 0 . 06 These are not claims of a “probabilistic tendency” in the current situation (but might be learned from past experience of similar situations) Probabilities of propositions change with new evidence: e.g., P ( A 25 | no reported accidents , 5 a.m. ) = 0 . 15 (Analogous to logical entailment status KB | = α , not truth.) Chapter 13 5
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Making decisions under uncertainty Suppose I believe the following: P ( A 25 gets me there on time | . . . ) = 0 . 04 P ( A 90 gets me there on time | . . . ) = 0 . 70 P ( A 120 gets me there on time | . . . ) = 0 . 95 P ( A 1440 gets me there on time | . . . ) = 0 . 9999 Which action to choose? Depends on my preferences for missing flight vs. airport cuisine, etc. Utility theory is used to represent and infer preferences Decision theory = utility theory + probability theory Chapter 13 6
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Probability basics Begin with a set Ω —the sample space e.g., 6 possible rolls of a die. ω Ω is a sample point / possible world / atomic event A probability space or probability model is a sample space with an assignment P ( ω ) for every ω Ω s.t. 0
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