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Probabilistic Inference Reading: Chapter 13 Next time: How should we define artificial intelligence? Reading for next time (see Links, Reading for Retrospective Class): Turing paper Mind, Brain and Behavior, John Searle Prepare discussion points by midnight, wed night (see end of slides)

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2 Transition to empirical AI Add in Ability to infer new facts from old Ability to generalize Ability to learn based on past observation Key: Observation of the world Best decision given what is known

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3 Overview of Probabilistic Inference Some terminology Inference by enumeration Bayesian Networks

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9 Probability Basics Sample space Atomic event Probability model An event A

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11 Random Variables Random variable Probability for a random variable

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17 Logical Propositions and Probability Proposition = event (set of sample points) Given Boolean random variables A and B: Event a = set of sample points where A(ω)=true Event ⌐ a=set of sample points where A(ω)=false Event aΛb=points where A(ω)=true and B(ω)=true Often the sample space is the Cartesian product of the range of variables Proposition=disjunction of atomic events in which it is true (aVb) = ( ⌐ aΛb)V(aΛ ⌐ b)V(aΛb) P(aVb)= P( ⌐ aΛb)+P(aΛ ⌐ b)+P(aΛb)

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25 Axioms of Probability All probabilities are between 0 and 1 Necessarily true propositions have probability 1. Necessarily false propositions have probability 0 The probability of a disjunction is P(aVb)=P(a)+P(b)-P(aΛb) P( ⌐ a)=1-p(a)

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26 The definitions imply that certain logically related events must have related probabilities P(aVb)= P(a)+P(b)-P(aΛb)

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27 Prior Probability Prior or unconditional probabilities of propositions P(female=true)=.5 corresponds to belief prior to arrival of any new evidence Probability distribution gives values for all possible assignments P(color) = (color = green, color=blue, color=purple) P(color)= (normalized: sums to 1) Joint probability distribution for a set of r.v.s gives the probability of every atomic event on those r.v.s (i.e., every sample point) P(color,gender) = a 3X2 matrix

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34 Inference by enumeration Start with the joint distribution

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35 Inference by enumeration P(HasTeeth)=.06+.12+.02=.2

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36 Inference by enumeration P(HasTeethVColor=Green)=.06+.12+.02+.24=.4 4

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37 Conditional Probability Conditional or posterior probabilities E.g., P(PlayerWins|HostOpenDoor=1 and PlayerPickDoor2 and Door1=goat) =.5 If we know more (e.g., HostOpenDoor=3 and door3-goat): P(PlayerWins)=1 Note: the less specific belief remains valid after more evidence arrives, but is not always useful New evidence may be irrelevant, allowing simplification: P(PlayerWins|California- earthquake)=P(PlayerWins)=.3

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38 Conditional Probability A general version holds for joint distributions: P(PlayerWins,HostOpensDoor1)=P(PlayerWins|HostOpensDoor1)*P(Ho stOpensDoor1)

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39 Inference by enumeration Compute conditional probabilities: P( ⌐Hasteeth|color=green)= P(⌐HasteethΛcolor=green) P(color=green) 0.8 = 0.24 0.06+.24

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40 Normalization Denominator can be viewed as normalization constraint α P( ⌐Hasteeth|color=green ) = α P( ⌐Hasteeth|color=green ) =α[P( ⌐Hasteeth,color=green, female )+ P( ⌐Hasteeth,color=green, ⌐ female)] =α[ + ]=α = Compute distribution on query variable by fixing evidence variables and summing over hidden variables

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41 Inference by enumeration

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42 Independence A and B are independent iff P(A|B)=P(A) or P(B|A)=P(B) or P(A,B)=P(A)P(B) 32 entries reduced to 12; for n independent biased coins, 2 n -> n Absolute independence powerful but rare Any domain is large with hundreds of variables none of which are independent

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44 Conditional Independence If I have length <=.2, the probability that I am female doesn’t depend on whether or not I have teeth: P(female|length<=.2,hasteeth)=P(female|h asteeth) The same independence holds if I am >.2 P(male|length>.2,hasteeth)=P(male|length>.2) Gender is conditionally independent of hasteeth given length

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45 In most cases, the use of conditional independence reduces the size of the representation of the joint distribution from exponential in n to linear in n Conditional independence is our most basic and robust form of knowledge about uncertain environments

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46 Next Class: Turing Paper A discussion class Graduate students and non-degree students: Anyone beyond a bachelor’s: Prepare a short statement on the paper. Can be your reaction, your position, a place where you disagree, an explication of a point. Undergraduates: Be prepared with questions for the graduate students All: Submit your statement or your question by midnight Wed night. All statements and questions will be printed and distributed in class on Wednesday.

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Probability: Review The state of the world is described using random variables Probabilities are defined over events –Sets of world states characterized.

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