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In this chapter, we explore the foundations of artificial intelligence through the lens of probabilistic inference. We discuss how to define AI, emphasizing key abilities such as inference of new facts, generalization, and learning from past observations. The chapter includes critical concepts like Bayesian networks, joint and conditional probabilities, and the axioms of probability. Students are encouraged to prepare discussion points based on the Turing paper, focusing on arguments and questions to deepen their understanding of AI theory and its empirical implications.
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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)
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
Overview of Probabilistic Inference • Some terminology • Inference by enumeration • Bayesian Networks
Probability Basics • Sample space • Atomic event • Probability model • An event A
Random Variables • Random variable • Probability for a random variable
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)
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)
The definitions imply that certain logically related events must have related probabilitiesP(aVb)= P(a)+P(b)-P(aΛb)
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)=<.6,.3,.1> (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
Inference by enumeration • Start with the joint distribution
Inference by enumeration • P(HasTeeth)=.06+.12+.02=.2
Inference by enumeration • P(HasTeethVColor=Green)=.06+.12+.02+.24=.44
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)=1Note: 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
Conditional Probability A general version holds for joint distributions: P(PlayerWins,HostOpensDoor1)=P(PlayerWins|HostOpensDoor1)*P(HostOpensDoor1)
Inference by enumeration • Compute conditional probabilities: • P(⌐Hasteeth|color=green)= P(⌐HasteethΛcolor=green) P(color=green)0.8 = 0.24 0.06+.24
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[<0.03,0.12>+<0.03,0.012>]=α<0.06,0.24>=<0.2,0.8> • Compute distribution on query variable by fixing evidence variablesand summing over hidden variables
Independence • A and B are independent iffP(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, 2n -> n • Absolute independence powerful but rare • Any domain is large with hundreds of variables none of which are independent
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|hasteeth) • 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
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
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.
