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Lecture 28 of 41 Uncertainty and Probabilistic Reasoning: Graphical Models Preliminaries Friday, 22 October 2004 William H. Hsu Department of Computing and Information Sciences, KSU http://www.kddresearch.org http://www.cis.ksu.edu/~bhsu Reading: Today: Chapter 13, Russell and Norvig 2e Friday and Next Week: Chapter 14 CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences Graphical Models of Probability • • 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 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 • Markov Condition for BBNs (Chain Rule): • Example BBN n P X 1 , X 2 , , X n P X i | parents X i i 1 Exposure-To-Toxins Age X1 X3 Cancer Serum Calcium X6 X5 Gender X2 X4 Smoking X7 Tumor Lung Descendants NonDescendants Parents P(20s, Female, Low, Non-Smoker, No-Cancer, Negative, Negative) = P(T) · P(F) · P(L | T) · P(N | T, F) · P(N | L, N) · P(N | N) · P(N | N) CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences Automated Reasoning using Probabilistic Models: Inference Tasks Adapted from slides by S. Russell, UC Berkeley CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences Semantics of Bayesian Networks Adapted from slides by S. Russell, UC Berkeley CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences Markov Blanket Adapted from slides by S. Russell, UC Berkeley CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences Constructing Bayesian Networks: The Chain Rule of Inference Adapted from slides by S. Russell, UC Berkeley CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences Example: Evidential Reasoning for Car Diagnosis Adapted from slides by S. Russell, UC Berkeley CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences BNJ Core [1] Design CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences BNJ Core [2] Graph Architecture © 2004 KSU BNJ Development Team CIS 730: Introduction to Artificial Intelligence CPCS-54 Network Kansas State University Department of Computing and Information Sciences BNJ Graphical User Interface: Network ALARM Network © 2004 KSU BNJ Development Team CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences BNJ Visualization [1] Framework © 2004 KSU BNJ Development Team CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences BNJ Visualization [2] Pseudo-Code Annotation (Code Page) ALARM Network © 2004 KSU BNJ Development Team CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences BNJ Visualization [3] Network Poker Network © 2004 KSU BNJ Development Team CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences Current Work: Features in Progress • Scalability – Large networks (50+ vertices, 10+ parents) – Very large data sets (106+) • Other Visualizations – K2 for structure learning – Conditioning • BNJ v1-2 ports – Guo’s dissertation algorithms – Importance sampling (CABeN) • Lazy Evaluation Barley Network CIS 730: Introduction to Artificial Intelligence © 2004 KSU BNJ Development Team Kansas State University Department of Computing and Information Sciences Terminology • Introduction to Reasoning under Uncertainty – Probability foundations – Definitions: subjectivist, frequentist, logicist – (3) Kolmogorov axioms • Bayes’s Theorem – Prior probability of an event – Joint probability of an event – Conditional (posterior) probability of an event • Maximum A Posteriori (MAP) and Maximum Likelihood (ML) Hypotheses – MAP hypothesis: highest conditional probability given observations (data) – ML: highest likelihood of generating the observed data – ML estimation (MLE): estimating parameters to find ML hypothesis • Bayesian Inference: Computing Conditional Probabilities (CPs) in A Model • Bayesian Learning: Searching Model (Hypothesis) Space using CPs CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences Summary Points • Introduction to Probabilistic Reasoning – Framework: using probabilistic criteria to search H – Probability foundations • Definitions: subjectivist, objectivist; Bayesian, frequentist, logicist • Kolmogorov axioms • Bayes’s Theorem – Definition of conditional (posterior) probability – Product rule • Maximum A Posteriori (MAP) and Maximum Likelihood (ML) Hypotheses – Bayes’s Rule and MAP – Uniform priors: allow use of MLE to generate MAP hypotheses – Relation to version spaces, candidate elimination • Next Week: Chapter 14, Russell and Norvig – Later: Bayesian learning: MDL, BOC, Gibbs, Simple (Naïve) Bayes – Categorizing text and documents, other applications CIS 730: Introduction to Artificial Intelligence Kansas State University Department of Computing and Information Sciences