Lecture 31 of 42 Reasoning under Uncertainty Uncertain

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Lecture 31 of 42 Reasoning under Uncertainty: Uncertain Inference Concluded Discussion: Fuzzy Reasoning &

Lecture 31 of 42 Reasoning under Uncertainty: Uncertain Inference Concluded Discussion: Fuzzy Reasoning & D-S Theory William H. Hsu Department of Computing and Information Sciences, KSU KSOL course page: http: //snipurl. com/v 9 v 3 Course web site: http: //www. kddresearch. org/Courses/CIS 730 Instructor home page: http: //www. cis. ksu. edu/~bhsu Reading for Next Class: Review Chapters 13 – 14, R&N Dempster-Shafer theory: http: //en. wikipedia. org/wiki/Dempster-Shafer_theory Fuzzy logic: http: //en. wikipedia. org/wiki/Fuzzy_logic CIS 530 / 730 Artificial Intelligence Lecture 31 of 42 Computing & Information Sciences Kansas State University

Lecture Outline l Reading for Next Class: Sections 14. 1 – 14. 2 (p.

Lecture Outline l Reading for Next Class: Sections 14. 1 – 14. 2 (p. 492 – 499), R&N 2 e l Last Class: Uncertainty, Probability, 13 (p. 462 – 486), R&N 2 e Where uncertainty is encountered: reasoning, planning, learning (later) Sources: sensor error, incomplete/inaccurate domain theory, randomness l Today: Probability Intro, Continued, Chapter 13, R&N 2 e Why probability ð Axiomatic basis: Kolmogorov ð With utility theory: sound foundation of rational decision making Joint probability Independence Probabilistic reasoning: inference by enumeration Conditioning ð Bayes’s theorem (aka Bayes’ rule) ð Conditional independence l Coming Week: More Applied Probability, Graphical Models CIS 530 / 730 Artificial Intelligence Lecture 31 of 42 Computing & Information Sciences Kansas State University

Acknowledgements Stuart J. Russell © 2004 -2005 Professor of Computer Science Chair, Department of

Acknowledgements Stuart J. Russell © 2004 -2005 Professor of Computer Science Chair, Department of Electrical Engineering and Computer Sciences Smith-Zadeh Prof. in Engineering University of California - Berkeley Russell, S. J. University of California, Berkeley http: //www. eecs. berkeley. edu/~russell / Peter Norvig, P. http: //norvig. com/ Director of Research Google Slides from: http: //aima. eecs. berkeley. edu Lotfali Asker-Zadeh (Lotfi A. Zadeh) Professor of Computer Science Department of Electrical Engineering and Computer Sciences Director, Berkeley Initiative in Soft Computing University of California - Berkeley CIS 530 / 730 Artificial Intelligence Lecture 31 of 42 © 2008 Zadeh, L. A. University of California, Berkeley http: //bit. ly/39 sh. SQ Computing & Information Sciences Kansas State University

Probability: Basic Definitions and Axioms l Sample Space ( ): Range of Random Variable

Probability: Basic Definitions and Axioms l Sample Space ( ): Range of Random Variable X l Probability Measure Pr( ) denotes range of observations; X: Probability Pr, or P: measure over power set 2 - event space In general sense, Pr(X = x ) is measure of belief in X = x ð P(X = x) = 0 or P(X = x) = 1: plain (aka categorical) beliefs ð Can’t be revised; all other beliefs are subject to revision l Kolmogorov Axioms 1. x . 0 P(X = x) 1 2. P( ) x P(X = x) = 1 3. l Joint Probability: P(X 1 X 2) Prob. of Joint Event X 1 X 2 l Independence: P(X 1 X 2) = P(X 1) P(X 2) CIS 530 / 730 Artificial Intelligence Lecture 31 of 42 Computing & Information Sciences Kansas State University

Evidential Reasoning – Inference by Enumeration Approach Based on slide © 2004 S. Russell

Evidential Reasoning – Inference by Enumeration Approach Based on slide © 2004 S. Russell & P. Norvig. Reused with permission. CIS 530 / 730 Artificial Intelligence Lecture 31 of 42 Computing & Information Sciences Kansas State University

Bayes’s Theorem (aka Bayes’ Rule) Based on slide © 2004 S. Russell & P.

Bayes’s Theorem (aka Bayes’ Rule) Based on slide © 2004 S. Russell & P. Norvig. Reused with permission. CIS 530 / 730 Artificial Intelligence Lecture 31 of 42 Computing & Information Sciences Kansas State University

Bayes’ Rule & Conditional Independence © 2004 S. Russell & P. Norvig. Reused with

Bayes’ Rule & Conditional Independence © 2004 S. Russell & P. Norvig. Reused with permission. CIS 530 / 730 Artificial Intelligence Lecture 31 of 42 Computing & Information Sciences Kansas State University

Non-Probabilistic Representation [1]: Concept of Fuzzy Set class unprecisiated precisiation generalization fuzzy set Informally,

Non-Probabilistic Representation [1]: Concept of Fuzzy Set class unprecisiated precisiation generalization fuzzy set Informally, a fuzzy set, A, in a universe of discourse, U, is a class with a fuzzy boundary. Adapted from slide © 2008 L. A. Zadeh, UC Berkeley CIS 530 / 730 Artificial Intelligence Lecture 31 of 42 http: //bit. ly/39 sh. SQ Computing & Information Sciences Kansas State University

Non-Probabilistic Representation [2]: Precisiation & Degree of Membership l Set A in U: Class

Non-Probabilistic Representation [2]: Precisiation & Degree of Membership l Set A in U: Class with Crisp Boundary l Precisiation: Association with Function whose Domain is U l Precisiation of Crisp Sets Through association with (Boolean-valued) characteristic function c. A: U {0, 1} l Precisiation of Fuzzy Sets Through association with membership function µA: U [0, 1] µA(u), u U, represents grade of membership of u in A l Degree of Membership in A: matter of degree “In fuzzy logic everything is or is allowed to be a matter of degree. ” – Zadeh Adapted from slide © 2008 L. A. Zadeh, UC Berkeley CIS 530 / 730 Artificial Intelligence Lecture 31 of 42 http: //bit. ly/39 sh. SQ Computing & Information Sciences Kansas State University

Non-Probabilistic Representation [3]: Fuzzy Set Example – Middle-Age l “Linguistic” Variables: Qualitative, Based on

Non-Probabilistic Representation [3]: Fuzzy Set Example – Middle-Age l “Linguistic” Variables: Qualitative, Based on Descriptive Terms l Imprecision of Meaning = Elasticity of Meaning l Elasticity of Meaning = Fuzziness of Meaning middle-age μ 1 0. 8 core of middle-age 0 45 40 55 60 43 definitely not middle-age definitely middle-age Adapted from slide © 2008 L. A. Zadeh, UC Berkeley CIS 530 / 730 Artificial Intelligence Lecture 31 of 42 definitely not middle-age http: //bit. ly/39 sh. SQ Computing & Information Sciences Kansas State University

Automated Reasoning using Probability: Inference Tasks Adapted from slide © 2004 S. Russell &

Automated Reasoning using Probability: Inference Tasks Adapted from slide © 2004 S. Russell & P. Norvig. Reused with permission. CIS 530 / 730 Artificial Intelligence Lecture 31 of 42 Computing & Information Sciences Kansas State University

Choosing Hypotheses l Bayes’s Theorem l MAP Hypothesis Generally want most probable hypothesis given

Choosing Hypotheses l Bayes’s Theorem l MAP Hypothesis Generally want most probable hypothesis given training data Define: value of x in sample space with highest f(x) Maximum a posteriori hypothesis, h. MAP l ML Hypothesis Assume that p(hi) = p(hj) for all pairs i, j (uniform priors, i. e. , PH ~ Uniform) Can further simplify and choose maximum likelihood hypothesis, h. ML CIS 530 / 730 Artificial Intelligence Lecture 31 of 42 Computing & Information Sciences Kansas State University

Graphical Models of Probability l Conditional Independence X is conditionally independent (CI) from Y

Graphical Models of Probability l Conditional Independence X is conditionally independent (CI) from Y given Z iff P(X | Y, Z) = P(X | Z) for all values of X, Y, and Z Example: P(Thunder | Rain, Lightning) = P(Thunder | Lightning) T R | L l Bayesian (Belief) Network Acyclic directed graph model B = (V, E, ) representing CI assertions over Vertices (nodes) V: denote events (each a random variable) Edges (arcs, links) E: denote conditional dependencies l Markov Condition for BBNs (Chain Rule): l Example BBN Exposure-To-Toxins Age X 1 X 3 Cancer Serum Calcium X 6 X 5 Gender X 2 X 4 X 7 Smoking Lung Tumor P(20 s, Female, Low, Non-Smoker, No-Cancer, Negative) = P(T) · P(F) · P(L | T) · P(N | T, F) · P(N | L, N) · P(N | N) CIS 530 / 730 Artificial Intelligence Lecture 31 of 42 Computing & Information Sciences Kansas State University

Evidential Reasoning: Example – Car Diagnosis Adapted from slide © 2004 S. Russell &

Evidential Reasoning: Example – Car Diagnosis Adapted from slide © 2004 S. Russell & P. Norvig. Reused with permission. CIS 530 / 730 Artificial Intelligence Lecture 31 of 42 Computing & Information Sciences Kansas State University

Tools for Building Graphical Models l Commercial Tools: Ergo, Netica, TETRAD, Hugin l Bayes

Tools for Building Graphical Models l Commercial Tools: Ergo, Netica, TETRAD, Hugin l Bayes Net Toolbox (BNT) – Murphy (1997 -present) Distribution page http: //http. cs. berkeley. edu/~murphyk/Bayes/bnt. html Development group http: //groups. yahoo. com/group/Bayes. Net. Toolbox l Bayesian Network tools in Java (BNJ) – Hsu et al. (1999 -present) Distribution page http: //bnj. sourceforge. net Development group http: //groups. yahoo. com/group/bndev Current (re)implementation projects for KSU KDD Lab • Continuous state: Minka (2002) – Hsu, Guo, Li • Formats: XML BNIF (MSBN), Netica – Barber, Guo • Space-efficient DBN inference – Meyer • Bounded cutset conditioning – Chandak CIS 530 / 730 Artificial Intelligence Lecture 31 of 42 Computing & Information Sciences Kansas State University

References: Graphical Models & Inference l Graphical Models Bayesian (Belief) Networks tutorial – Murphy

References: Graphical Models & Inference l Graphical Models Bayesian (Belief) Networks tutorial – Murphy (2001) http: //www. cs. berkeley. edu/~murphyk/Bayes/bayes. html Learning Bayesian Networks – Heckerman (1996, 1999) http: //research. microsoft. com/~heckerman l Inference Algorithms Junction Tree (Join Tree, L-S, Hugin): Lauritzen & Spiegelhalter (1988) http: //citeseer. nj. nec. com/huang 94 inference. html (Bounded) Loop Cutset Conditioning: Horvitz & Cooper (1989) http: //citeseer. nj. nec. com/shachter 94 global. html Variable Elimination (Bucket Elimination, Elim. Bel): Dechter (1986) http: //citeseer. nj. nec. com/dechter 96 bucket. html Recommended Books • Neapolitan (1990) – out of print; see Pearl (1988), Jensen (2001) • Castillo, Gutierrez, Hadi (1997) • Cowell, Dawid, Lauritzen, Spiegelhalter (1999) Stochastic Approximation http: //citeseer. nj. nec. com/cheng 00 aisbn. html CIS 530 / 730 Artificial Intelligence Lecture 31 of 42 Computing & Information Sciences Kansas State University

Terminology l Uncertain Reasoning: Inference Task with Uncertain Premises, Rules l Probabilistic Representation Views

Terminology l Uncertain Reasoning: Inference Task with Uncertain Premises, Rules l Probabilistic Representation Views of probability ð Subjectivist: measure of belief in sentences ð Frequentist: likelihood ratio ð Logicist: counting evidence Founded on Kolmogorov axioms ð Sum rule ð Prior, joint vs. conditional ð Bayes’s theorem & product rule: P(A | B) = (P(B | A) * P(A)) / P(B) Independence & conditional independence l Probabilistic Reasoning Inference by enumeration Evidential reasoning CIS 530 / 730 Artificial Intelligence Lecture 31 of 42 Computing & Information Sciences Kansas State University

Summary Points l Last Class: Reasoning under Uncertainty and Probability (Ch. 13) Uncertainty is

Summary Points l Last Class: Reasoning under Uncertainty and Probability (Ch. 13) Uncertainty is pervasive What are we uncertain about? l Today: Chapter 13 Concluded, Preview of Chapter 14 Why probability ð Axiomatic basis: Kolmogorov ð With utility theory: sound foundation of rational decision making Joint probability Independence Probabilistic reasoning: inference by enumeration Conditioning ð Bayes’s theorem (aka Bayes’ rule) ð Conditional independence l Coming Week: More Applied Probability Graphical models as KR for uncertainty: Bayesian networks, etc. Some inference algorithms for Bayes nets CIS 530 / 730 Artificial Intelligence Lecture 31 of 42 Computing & Information Sciences Kansas State University

Chapter 9 Reasoning in Uncertain Situations Contents UncertainChapter 9 Reasoning in Uncertain Situations Contents Uncertain
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DecisionMaking under Uncertainty ONR MURI Uncertain Multiagent SystemsDecisionMaking under Uncertainty ONR MURI Uncertain Multiagent Systems
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Uncertainty Uncertainty n n n Uncertainty is aUncertainty Uncertainty n n n Uncertainty is a
Reasoning with Uncertainty Probabilistic Reasoning Foundations of ArtificialReasoning with Uncertainty Probabilistic Reasoning Foundations of Artificial
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Patterns and Inductive Reasoning Inductive reasoning is reasoningPatterns and Inductive Reasoning Inductive reasoning is reasoning
CaseBased Reasoning CaseBased Reasoning CBR Casebased reasoning CBRCaseBased Reasoning CaseBased Reasoning CBR Casebased reasoning CBR
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