Quantifying Uncertainty - PDF Free Download

Quantifying Uncertainty CSL 302 ARTIFICIAL INTELLIGENCE SPRING 2014

Outline Probability theory - basics Probabilistic Inference oinference by enumeration Conditional Independence Bayesian Networks Learning Probabilistic Models onaïve Bayes Classifier CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 2

Non-monotonic logic Traditional logic is monotonic othe set of legal conclusions grows monotonically with the set of facts appearing in our initial database When humans reason, we use defeasible logic oalmost every conclusion we draw is subject to reversal oif we find contradicting information later, we ll want to retract earlier inferences Nonmonotonic logic, or defeasible reasoning, allows a statement to be retracted Solution: Truth Maintenance okeep explicit information about which facts/inferences support other inferences oif the foundation disappears, so must the conclusion CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 3

Uncertainty On the other hand, the problem might not be in the fact that T/F values can change over time but rather that we are not certain of the T/F value Agents almost never have access to the whole truth about their environment Agents must act in the presence of uncertainty oincompleteness and/or incorrectness of rules used by the agent olimited and ambiguous sensors oimperfection/noise in agent s actions odynamic nature of the environment CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 4

Pitfalls of pure logic Laziness otoo much work to list all conditions needed to ensure an exceptionless rule Theoretical ignorance oscience has no complete theory for the domain Practical ignorance oeven if we know all the rules, we may be uncertain about them owe only have a degree of belief in them CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 5

Probability Theory vs. Logic Probability - tool for handling degrees of belief osummarizes uncertainty due to laziness and ignorance Logic Probability Theory Ontological commitments world is composed of facts that do or do not hold world is composed of facts that do or do not hold Epistemological commitments each sentence is true or false or unknown numerical degree of belief between 0(sentence for certainly false) and 1(sentences are certainly true) CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 6

Rational Agent Approach Choose action A that maximizes expected utility omaximizes Prob(A)*Utility(A) Prob(A) - probability that A will succeed Utility(A) - utility value to agent of A s outcomes CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 7

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Global semantics Global semantics defines the full joint distribution as the product of the local conditional distributions: P(x 1,..., x n ) = Π n i = 1P(x i par ents(x i )) e.g., P(j m a b e) = B J A E M CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 35

Global semantics Global semantics defines the full joint distribution as the product of the local conditional distributions: B E P(x 1,..., x n ) = Π n i = 1P(x i par ents(x i )) A e.g., P(j m a b e) = P(j a)p(m a)p(a b, e)p( b)p( e) = 0.9 0.7 0.001 0.999 0.998 0.00063 J M CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 36

Const r uct ing Bayesian net wor ks Need a method such that a series of locally testable assertions of conditional independence guarantees the required global semantics 1. Choose an ordering of variables X 1,..., X n 2. For i = 1 to n add X i to the network select parents from X 1,..., X i 1 such that P(X i Par ents(x i )) = P(X i X 1,..., X i 1 ) This choice of parents guarantees the global semantics: P(X 1,..., X n ) = Π n i = 1P(X i X 1,..., X i 1 ) (chain rule) = Π n i = 1 P(X i Par ents(x i )) (by const ruction) CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 37

Example Suppose we choose the ordering M, J, A, B, E MaryCalls JohnCalls P(J M ) = P(J )? CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 38

Example Suppose we choose the ordering M, J, A, B, E MaryCalls JohnCalls Alarm P(J M ) = P(J )? No P(A J, M ) = P(A J )? P(A J, M ) = P(A)? CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 39

Example Suppose we choose the ordering M, J, A, B, E MaryCalls JohnCalls Alarm Burglary P(J M ) = P(J )? No P(A J, M ) = P(A J )? P(A J, M ) = P(A)? P(B A,J, M ) = P(B A)? P(B A,J, M ) = P(B )? No CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 40

Example Suppose we choose the ordering M, J, A, B, E MaryCalls JohnCalls Alarm Burglary Earthquake P(J M ) = P(J )? No P(A J, M ) = P(A J )? P(A J, M ) = P(A)? P(B A,J, M ) = P(B A)? Yes P(B A,J, M ) = P(B )? No P(E B, A,J, M ) = P(E A)? P(E B, A,J, M ) = P(E A,B )? No CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 41 Chapt er 14.1 3 16

Example Suppose we choose the ordering M, J, A, B, E MaryCalls JohnCalls Alarm Burglary Earthquake P(J M ) = P(J )? No P(A J, M ) = P(A J )? P(A J, M ) = P(A)? P(B A,J, M ) = P(B A)? Yes P(B A,J, M ) = P(B )? No P(E B, A,J, M ) = P(E A)? No P(E B, A,J, M ) = P(E A,B )? Yes No CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 42 Chapt er 14.1 3 17

Example cont d. MaryCalls JohnCalls Alarm Burglary Earthquake Deciding conditional independence is hard in noncausal directions (Causal models and conditional independence seem hardwired for humans!) Assessing conditional probabilities is hard in noncausal directions Network is less compact: 1+ 2+ 4+ 2+ 4= 13 numbers needed CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 43 Chapt er 14.1 3 18

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I nfer ence by enumer at ion Slightly intelligent way to sum out variables from the joint without actually constructing its explicit representation Simple query on the burglary network: P(B j,m) = P(B, j,m)/p(j,m) = αp(b, j,m) = α Σ e Σ a P(B, e, a, j,m) B J A E M Rewrite full joint entries using product of CPT entries: P(B j,m) = α Σ e Σ a P(B )P(e)P(a B, e)p(j a)p(m a) = αp(b ) Σ e P(e) Σ a P(a B, e)p(j a)p(m a) Recursive depth-first enumeration: O(n) space, O(d n ) time CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 45

Enumer at ion algor it hm funct ion Enumer at ion-a sk (X,e,bn) returns a distribution over X input s: X, the query variable e, observed values for variables E bn, a Bayesian network with variables { X } E Y Q(X ) a distribution over X, initially empty for each value x i of X do extend e with value x i for X Q(x i ) Enumer at e-al l (Var s[bn],e) return Nor mal ize(q(x )) funct ion Enumer at e-al l (vars,e) returns a real number if Empt y?(vars) then return 1.0 Y Fir st (vars) if Y has value y in e then return P(y Pa(Y )) Enumer at e-al l (Rest (vars),e) else return y P(y Pa(Y )) Enumer at e-al l (Rest (vars),e y ) where e y is e extended with Y = y CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 46 Chapt er 14.4 5 5

Evaluation tree P(b).001 P(e).002 P( e).998 P(a b,e) P( a b,e) P(a b, e) P( a b, e).95.05.94.06 P(j a).90 P(j a).05 P(j a).90 P(j a).05 P(m a) P(m a).70.01 P(m a) P(m a).70.01 Enumeration is inefficient: repeated computation e.g., computes P(j a)p(m a) for each value of e CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 47

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A B C P(A,B,C) T T T 0.1 T T F 0.1 T F T 0.1 T F F 0.1 F T T 0.1 F T F 0.1 F F T 0.1 F F F 0.3 CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 49

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14.4. Exact Inference in Bayesian Networks 527 T F.7 T F.8 T T F.3 F T.9 F T.6 T F T.7 F F.1 F F.4 T F F.7 A B f 1 (A,B ) B C f 2 (B,C) A B C f 3 (A,B, C) T T.3 T T.2 T T FT T.3.2= T.06.9 T F.7 T F.8 T T F.3.8=.24 F T F.9 F T.9 F T.6 T F T.7.6=.42 F F.1 F F.4 T F F F F.7.4= T.28.1 F T FT F.9.2= F.18.1 F T F.9.8=.72 Figure 14.10 Illustrating pointwise multiplication: F F f 1 (A,B T ) f.1 2 (B,C).6=.06 = f 3 (A,B F F F.1.4=.04 Summing out a variable from a product of factors is done by adding up the su Figure 14.10 Illustrating pointwise multiplication: f 1 (A,B) f 2 (B,C) = f 3 (A,B, C). formed by fixing the variable to each of its values in turn. For example, to sum o f 3 (A,B, C),wewrite Summing out a variable from a product of factors is done by adding up the submatrices formed by fixing the variable to each of its values in turn. For example, to sum out A from f(b,c) = f f 3 (A,B, C), we write 3 (A,B, C) = f 3 (a, B, C) + f 3 ( a, B, C) f(b,c) = = = a a f 3 (A,B, C) = f 3 (a, B, C) + f 3 ( a, B, C).06.24.42.28.06.24.42.28 + +.18.72.06.04.18.72.06.04 =.24.96.48.32 The only trick is to notice that any factor that does not depend on the variable to be summed out burglary can be moved network, outside the the relevant summation. part of Forthe example, expression if we were would to sum be out E first in the burglary network, the relevant part of the expression would be CSL 302 ARTIFICIAL INTELLIGENCE, INDIAN INSTITUTE OF TECHNOLOGY ROPAR 55 f (E ) f (A,B, E) f (A) f (A) = f (A) f (A) f (E ) f (A,B, E). =.24.96.48.32 The only trick is to notice that any factor that does not depend on the variable to b out can be moved outside the summation. For example, if we were to sum out E f 2 (E ) f 3 (A,B, E) f 4 (A) f 5 (A) = f 4 (A) f 5 (A).. f 2 (E) f

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