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Inference using Graphical Models and Software Tools

Lecture 31 of 42. Inference using Graphical Models and Software Tools. Monday, 07 November 2005 William H. Hsu Laboratory for Knowledge Discovery in Databases Department of Computing and Information Sciences Kansas State University http://www.kddresearch.org

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Inference using Graphical Models and Software Tools

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  1. Lecture 31 of 42 Inference using Graphical Modelsand Software Tools Monday, 07 November 2005 William H. Hsu Laboratory for Knowledge Discovery in Databases Department of Computing and Information Sciences Kansas State University http://www.kddresearch.org This presentation is based upon: http://www.kddresearch.org/KSU/CIS/Math-20021107.ppt

  2. 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 Exposure-To-Toxins Serum Calcium X6 X1 X3 Age Cancer X5 X2 X4 X7 Gender Smoking Lung Tumor Graphical Models Overview [1]:Bayesian Networks 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)

  3. Upward (child-to-parent)  messages C4 C5 C1 C2 C3 C6 ’ (Ci’) modified during  message-passing phase Downward  messages P’ (Ci’) is computed during  message-passing phase Propagation Algorithm in Singly-Connected Bayesian Networks – Pearl (1983) Multiply-connected case: exact, approximate inference are #P-complete (counting problem is #P-complete iff decision problem is NP-complete) Adapted from Neapolitan (1990), Guo (2000)

  4. Find Maximal Cliques Triangulate A1 Clq4 Clq1 F6 B2 Moralize Clq2 G5 E3 B2 E3 G5 G C4 G A H F B A1 D8 B2 E3 D F6 H7 D B E A E F H C C G5 E3 Bayesian Network (Acyclic Digraph) C4 G5 Clq3 C4 C4 Clq6 C4 D8 Clq5 H7 Inference by Clustering [1]: Graph Operations (Moralization, Triangulation, Maximal Cliques) Adapted from Neapolitan (1990), Guo (2000)

  5. Inference by Clustering [2]:Junction Tree – Lauritzen & Spiegelhalter (1988) • Input: list of cliques of triangulated, moralized graphGu • Output: • Tree of cliques • Separators nodes Si, • Residual nodes Ri and potential probability (Clqi) for all cliques • Algorithm: • 1. Si = Clqi(Clq1  Clq2 … Clqi-1) • 2. Ri = Clqi - Si • 3. If i >1 then identify a j < i such that Clqjis a parent of Clqi • 4. Assign each node v to a unique clique Clqi that v  c(v)  Clqi • 5. Compute (Clqi) = f(v) Clqi = P(v | c(v)) {1 if no v is assigned to Clqi} • 6. Store Clqi , Ri , Si, and (Clqi) at each vertex in the tree of cliques Adapted from Neapolitan (1990), Guo (2000)

  6. Ri: residual nodes Si: separator nodes (Clqi): potential probability of Clique i AB (Clq1) = P(B|A)P(A) Clq1 = {A, B} A1 Clq1 Clq4 R1 = {A, B} Clq1 S1 = {} F6 B B2 BEC Clq2 (Clq2) = P(C|B,E) G5 E3 Clq2 = {B,E,C} Clq2 B2 E3 R2 = {C,E} S2 = { B } G5 EC E3 ECG Clq3 = {E,C,G} C4 Clq3 R3 = {G} (Clq3) = 1 G5 S3 = { E,C } Clq3 C4 EG CG C4 EGF (Clq4) = P(E|F)P(G|F)P(F) CGH (Clq5) = P(H|C,G) Clq6 C4 Clq5 Clq4 (Clq2) = P(D|C) D8 Clq4 = {E, G, F} Clq5 = {C, G,H} R4 = {F} R5 = {H} Clq5 C H7 S4 = { E,G } S5 = { C,G } CD Clq6 = {C, D} Clq6 R5 = {D} S5 = { C} Inference by Clustering [3]:Clique-Tree Operations Adapted from Neapolitan (1990), Guo (2000)

  7. Age = [0, 10) X1,1 Age = [10, 20) X1,2 Age = [100, ) X1,10 Inference by Loop Cutset Conditioning • Deciding Optimal Cutset: NP-hard • Current Open Problems • Bounded cutset conditioning: ordering heuristics • Finding randomized algorithms for loop cutset optimization Split vertex in undirected cycle; condition upon each of its state values Exposure-To- Toxins Serum Calcium Number of network instantiations: Product of arity of nodes in minimal loop cutset Cancer X3 X6 X5 X4 X7 Smoking Lung Tumor X2 Gender Posterior: marginal conditioned upon cutset variable values

  8. Inference by Variable Elimination [1]:Intuition Adapted from slides by S. Russell, UC Berkeley http://aima.cs.berkeley.edu/

  9. Inference by Variable Elimination [2]:Factoring Operations Adapted from slides by S. Russell, UC Berkeley http://aima.cs.berkeley.edu/

  10. Season A Rain Sprinkler B C F D Wet Manual Watering G Slippery Inference by Variable Elimination [3]:Example P(A), P(B|A), P(C|A), P(D|B,A), P(F|B,C), P(G|F) P(G|F) G=1 P(D|B,A) P(F|B,C) P(B|A) P(C|A) P(A) P(A|G=1) = ? d = < A, C, B, F, D, G > λG(f) = ΣG=1 P(G|F) Adapted from Dechter (1996), Joehanes (2002)

  11. References [1]:Graphical Models and Inference Algorithms • 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 • Inference Algorithms • Junction Tree (Join Tree, L-S, Hugin): Lauritzen & Spiegelhalter (1988) http://citeseer.nj.nec.com/huang94inference.html • (Bounded) Loop Cutset Conditioning: Horvitz & Cooper (1989) http://citeseer.nj.nec.com/shachter94global.html • Variable Elimination (Bucket Elimination, ElimBel): Dechter (1986)http://citeseer.nj.nec.com/dechter96bucket.html • Recommended Books • Neapolitan (1990, 2003); see Pearl (1988), Jensen (2001) • Castillo, Gutierrez, Hadi (1997) • Cowell, Dawid, Lauritzen, Spiegelhalter (1999) • Stochastic Approximation http://citeseer.nj.nec.com/cheng00aisbn.html

  12. References [2]:Machine Learning, KDD, and Bioinformatics • Machine Learning, Data Mining, and Knowledge Discovery • K-State KDD Lab: literature survey and resource catalog (1999-present) http://www.kddresearch.org/Resources • Bayesian Network tools in Java (BNJ): Hsu, Barber, King, Meyer, Thornton (2002-present) http://bnj.sourceforge.net • Machine Learning in Java (BNJ): Hsu, Louis, Plummer (2002) http://mldev.sourceforge.net • Bioinformatics • European Bioinformatics Institute Tutorial: Brazma et al. (2001) http://www.ebi.ac.uk/microarray/biology_intro.htm • Hebrew University: Friedman, Pe’er, et al. (1999, 2000, 2002) http://www.cs.huji.ac.il/labs/compbio/ • K-State BMI Group: literature survey and resource catalog (2002-2005) http://www.kddresearch.org/Groups/Bioinformatics

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