Computing & Information Sciences Kansas State University Data Sciences Summer Institute Multimodal Information Access and Synthesis Learning and Reasoning with Graphical Models of Probability for the Identity Uncertainty Problem William H. Hsu Tuesday, 29 May 2007 Laboratory for Knowledge Discovery in Databases Kansas State University University of Illinois at Urbana-ChampaignDSSI--MIAS

Computing & Information Sciences Kansas State University Graphical Models of Probability –Markov graphs –Bayesian (belief) networks –Causal semantics –Direction-dependent separation (d-separation) property Learning and Reasoning: Problems, Algorithms –Inference: exact and approximate Junction tree – Lauritzen and Spiegelhalter (1988) (Bounded) loop cutset conditioning – Horvitz and Cooper (1989) Variable elimination – Dechter (1996) –Structure learning K2 algorithm – Cooper and Herskovits (1992) Variable ordering problem – Larannaga (1996), Hsu et al. (2002) Probabilistic Reasoning in Machine Learning, Data Mining Current Research and Open Problems Part 1 of 8: Graphical Models Intro Overview

Computing & Information Sciences Kansas State University Adapted from Fayyad, Piatetsky-Shapiro, and Smyth (1996) Stages of Data Mining

Computing & Information Sciences Kansas State University 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) Conditional Independence –X is conditionally independent (CI) from Y given Z (sometimes written X  Y | 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 X1X1 X3X3 X4X4 X5X5 Age Exposure-To-Toxins Smoking Cancer X6X6 Serum Calcium X2X2 Gender X7X7 Lung Tumor Graphical Models Defined [1]: Independence and Bayes Nets

Computing & Information Sciences Kansas State University Z XEY (1) (2) (3) Z Z From S. Russell & P. Norvig (1995) Adapted from J. Schlabach (1996) Motivation: The conditional independence status of nodes within a BBN might change as the availability of evidence E changes. Direction-dependent separation (d-separation) is a technique used to determine conditional independence of nodes as evidence changes. Definition: A set of evidence nodes E d-separates two sets of nodes X and Y if every undirected path from a node in X to a node in Y is blocked given E. A path is blocked if one of three conditions holds: Graphical Models Defined [2]: D-Separation and Markov Blankets

Computing & Information Sciences Kansas State University Adapted from slides by S. Russell, UC Berkeley Multiply-connected case: exact, approximate inference are # P -complete Graphical Models Defined [3]: Reasoning with Bayes Nets

Computing & Information Sciences Kansas State University Goal: Estimate Filtering: r = t –Intuition: infer current state from observations –Applications: signal identification –Variation: Viterbi algorithm Prediction: r < t –Intuition: infer future state –Applications: prognostics Smoothing: r > t –Intuition: infer past hidden state –Applications: signal enhancement CF Tasks –Plan recognition by smoothing –Prediction cf. WebCANVAS – Cadez et al. (2000) Adapted from Murphy (2001), Guo (2002) Bayesian Network Applications [1]: Time Series Prediction

Computing & Information Sciences Kansas State University General-Case BBN Structure Learning: Use Inference to Compute Scores Optimal Strategy: Bayesian Model Averaging –Assumption: models h  H are mutually exclusive and exhaustive –Combine predictions of models in proportion to marginal likelihood Compute conditional probability of hypothesis h given observed data D i.e., compute expectation over unknown h for unseen cases Let h  structure, parameters   CPTs Posterior ScoreMarginal Likelihood Prior over StructuresLikelihood Prior over Parameters Bayesian Network Applications [2]: Bayes Optimal Classification

Computing & Information Sciences Kansas State University Split vertex in undirected cycle; condition upon each of its state values Number of network instantiations: Product of arity of nodes in minimal loop cutset Posterior: marginal conditioned upon cutset variable values X3X3 X4X4 X5X5 Exposure-To- Toxins Smoking Cancer X6X6 Serum Calcium X2X2 Gender X7X7 Lung Tumor X 1,1 Age = [0, 10) X 1,2 Age = [10, 20) X 1,10 Age = [100,  ) Deciding Optimal Cutset: NP-hard Current Open Problems –Bounded cutset conditioning: ordering heuristics –Finding randomized algorithms for loop cutset optimization Inference in Bayesian Networks: Loop Cutset Conditioning

Computing & Information Sciences Kansas State University Novel Contributions [3]: Learning in Graphical Models Continuing Work: Speeding up Approximate Inference using Edge Deletion - J. Thornton (2005) Bayesian Network tools in Java (BNJ) v4 - W. Hsu, J. M. Barber, J. Thornton (2006) Dynamic Bayes Net for Prediction University of Illinois at Urbana-ChampaignDSSI--MIAS

Computing & Information Sciences Kansas State University © 2005 KSU Bayesian Network tools in Java (BNJ) Development Team ALARM Network Bayesian Network tools in Java (BNJ) v4 University of Illinois at Urbana-ChampaignDSSI--MIAS

Computing & Information Sciences Kansas State University Questions and Discussion University of Illinois at Urbana-ChampaignDSSI--MIAS