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1 Chapter 5 Belief Updating in Bayesian Networks Bayesian Networks and Decision Graphs Finn V. Jensen Qunyuan Zhang Division. of Statistical Genomics, CGS Statistical Genetics Forum May 7,2007

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2 Contents of the Book I A practical Guide to Normative Systems 1 Causal and Bayesian Network 2 Building Models 3 Learning, Adaption, and Tuning 4 Decision Graphs II Algorithms for Normative Systems 5 Belief Updating in Bayesian Network 6 Bayesian Network Analysis Tools 7 Algorithms for Influence Diagrams

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3 Structure of the Book 1 Causal and Bayesian Network 2 Building Models 3 Learning, Adaption, and Tuning 5 Belief Updating in Bayesian Network 6 Bayesian Network Analysis Tools 4 Decision Graphs 7 Algorithms for Influence Diagrams I. What is BN? II. How to create a BN? III. What can we use BN to do? and how? [to know sth.] Prob.(a single variable | BN) Joint Prob.(a set variables | BN) Importance of varibales evidence sensitivity parameter sensitivity Data conflict analysis [to make decision] Optimal decision (cost & gain)

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4 BN & Decision Tree A D1D1 V1V1 B T D2D2 V2V2 C U=V 1 +V 2 D1D1 T AXC V 1 +V 2 D2D2 AXC V 1 +V 2 AXC V 1 +V 2 D2D2 AXC V 1 +V 2 T AXC V 1 +V 2 D2D2 AXC V 1 +V 2 AXC V 1 +V 2 D2D2 AXC V 1 +V 2 P(A,C|D 1,T,D 2 )

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5 “BN” of the Book Concept of BN Model Biulding (known part of structure) BN Learning (uncertain part of structure) BN (structure & parameters) Rules & TheoriesData & Algorithms Probability Calculation Knowing, Understanding & Explaining Decisions Actions Cost & Gain Changes

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6 Chapter 5 Belief Updating in Bayesian Networks Belief = Probability Belief updating = Probability calculating based on a BN (model, parameters and/or evidences) Linear Model BN Logistic Model X1X2X3 e Y Conditional Probability P(Y| X1,X2,X3) Marginal Probability P(Y) =∑ [-Y] φ X2 X1 X3 Y C A B E D F

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7 Marginal Probability Calculation in BN I. Simplification (5.5) II. Marginalization (5.2),(5.3),(5.4),(5.6) III. Simulation (5.7)

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8 I. Simplifications Graph-theoretic Representation Definitions, Propositions & Theorems Barren Nodes D A B C F E G e D A B C F E G eG e D A B C F E D A B C F E e e D A B C F E e d-separation By excluding the non-informative nodes (white nodes)

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9 II. Marginalization Calculating sums of products of potentials by eliminating variables repeatedly

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10 Marginal Probabilities A B A1A2P(B) B1p1p2 p1+p2 P(B1) B2p3p4 p3+p4 P(B2) P(A) p1+p3 P(A1) p2+p4 P(A2) Joint Probabilities

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11 An Example of Marginalization/Elimination BN parameters (potentials) : φ 1 =P (A 1 ), φ 2 =P (A 2 |A 1 ), φ 3 =P (A 3 |A 1 ), φ 4 =P (A 4 |A 2 ) φ 5 =P (A 5 |A 2, A 3 ), φ 6 =P (A 6 |A 3 ) P(A 4 )=? A3 A1 A2 A4A5A6 Distributive Law

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12 Marginalization/Elimination Order A3 A1 A2 A4A5A6 Variable Elimination Order

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13 Marginalization/Elimination Graph-theoretic Representation Definitions, Propositions & Theorems Domain: a set of variables in BN Potential : a real-valued probabilistic table over a domain φ 1 =P (A 1 ), φ 2 =P (A 2 |A 1 ), φ 3 =P (A 3 |A 1 ), φ 4 =P (A 4 |A 2 ) φ 5 =P (A 5 |A 2, A 3 ), φ 6 =P (A 6 |A 3 ) A3 A1 A2 A4A5A6 Definition 5.1 (Elimination) Let Ф be a set of potentials, and let X be a variable. X is eliminated from Ф by: 1.Remove all potentials in Ф with X in their domains. Call the removed set Ф X X= A 3 => Ф X =(φ 3, φ 5, φ 6 ), Ф =(φ 1, φ 2, φ 4 ) 2.Calculate φ -X = ∑ x Π Ф X = ∑ A3 φ 3 φ 5 φ 6 3.Add φ -X to Ф. Call the result set Ф -X =(φ 1, φ 2, φ 4, φ -X ) P(Y) is calculated by repeatedly eliminating the variables except Y Question : how to find an efficient/optimal elimination order?

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14 Domain Graphs Graph-theoretic Representation Definitions, Propositions & Theorems BN graph 6 domains φ 1 (A 1 ), φ 2 (A 2,A 1 ), φ 3 (A 3,A 1 ), φ 4 (A 4,A 2 ) φ 5 (A 5,A 2,A 3 ), φ 6 (A 6,A 3 ) A3 A1 A2 A4A5A6 Domain graph 6 domains φ 1 (A 1 ), φ 2 (A 2,A 1 ), φ 3 (A 3,A 1 ), φ 4 (A 4,A 2 ) φ 5 (A 5,A 2,A 3 ), φ 6 (A 6,A 3 ) A3 A1 A2 A4A5A6

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15 Perfect Elimination Sequence Graph-theoretic Representation Definitions, Propositions & Theorems Fill-ins (red links) Perfect Elimination Sequence An elimination sequence without introducing fill-ins. e.g. A6, A5, A3, A1, A2 down to A4 => P(A4) A5, A6, A3, A1, A2 down to A4 => P(A4) A1, A5, A6, A3, A2 down to A4 => P(A4) A3 A1 A2 A4A5A6 A1 A2 A4A5A6

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16 Domain Set of Elimination Sequence Graph-theoretic Representation Definitions, Propositions & Theorems The domain set of an elimination sequence is the set of domains of potentials produced during the elimination where potentials that are subsets of other potentials are removed. For the sequence A6, A5, A3, A1, A2 down to A4 => P(A4) the set of domains is {(A6,A3),(A2,A3,A5),(A1,A2,A3), (A1,A2),(A2,A4)} Domain set reflects the complexity of an elimination sequence. Question: how to find the smallest domain set ?

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17 Set of Cliques Graph-theoretic Representation Definitions, Propositions & Theorems All perfect elimination sequences produce the same the domain set, namely the set of cliques of the domain graph. e.g. all the sequences A6, A5, A3, A1, A2 down to A4 A5, A6, A3, A1, A2 down to A4 A1, A5, A6, A3, A2 down to A4 produce the domain set {(A6,A3),(A2,A3,A5),(A1,A2,A3), (A1,A2),(A2,A4)} which contains 5 domains / cliques Any perfect elimination sequence is optimal. Cliques are a set of domains produce by perfect elimination sequences. Clique set is the optimal set of domains. Question: how to determine the set of cliques?

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18 Triangulated Graphs Graph-theoretic Representation Definitions, Propositions & Theorems An undirected graph with a perfect elimination sequence is called a triangulated graph. A triangulated graph A nontriangulated graph Perfect elimination sequence No perfect elimination sequence A5, A2, A4, A3 down to A1 A3 A1 A2 A4A5 A3 A1 A2 A4A5

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19 Cliques in Triangulated Graphs Graph-theoretic Representation Definitions, Propositions & Theorems X : a node in domain graph Fx : the set of neighbor nodes of X plus X Simplicial: nodes with a complete neighbor set are called simplicial To determine the set of cliques in a triangulated graph 1. Eliminate a simplicial node X. Fx is a clique candidate. 2. If Fx does not include all remaining nodes, go to 1. 3. Prune the set of cliques candidates by removing sets that are subsets of other clique candidates. 4. The resulting set is the set of cliques. Question: given a set of cliques, how to determine the perfect elimination order? D A B C E X

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20 Join Tree Graph-theoretic Representation Definitions, Propositions & Theorems An organized tree of cliques, in which all nodes on the path between V and W contain the intersection of V and W. D A B CF I E GH J ABCD V1 BCD S1 CGHJ V5 CG S5 BCDE V10 BCDG V1 BCD S1 DEFI V3 DE S3 ABCD CGHJ BCDE BCDG DEFIABCD CGHJ BCDE BCDG DEFI A domain graph Cliques (V) and Separators (S) A join tree Elimination sequence A,F,I,H,J,G,B,C,D down to E Not a join tree

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21 Propagation Junction Trees Graph-theoretic Representation Definitions, Propositions & Theorems A junction tree is a join tree with the following structure: 1. Each potential is attached to a clique containing the domain of this potential (cliques) 2. Each link has the appropriate separator attached (separable) 3. Each separator contains two “mailboxes”, one for each direction (mutual communication) φ 1,φ 2,φ 3 V4: A1, A2, A3 φ 4 V6: A2, A4 φ 5 V2: A2, A3, A5 φ 6 V1: A3, A6 ↑ ↓ S4:A2 ↑ ↓ S2:A2,A3 ↑ ↓ S1:A3 Collect evidence to V6 distribute evidence from V6 Junction trees provide a general framework for finding optimal elimination sequence for triangulated graphs. Question: what if a graph is non-triangulated?

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22 Triangulations Graph-theoretic Representation Definitions, Propositions & Theorems Convert a non-triangulated graph into a triangulated one by adding new link(s) BN non-triangulated graph triangulated graph D A BC E F G H IJ D A BC E F G H IJ D A BC E F G H IJ Optimal triangulation? Minimal fill-in size? Heuristic approach: eliminate repeatedly a smplicial node, and if this is not possible, eliminate a node X with minimal size of Fx.

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23 III. Stochastic Simulations Forward Sampling 1. P(A) => A 2. P(B|A)=>B, P(C|A)=>C 3. P(D|B)=>D 4. P(E|C,D)=>E 5. Repeat steps 1~4 D A B C E Gibbs Sampling Evidence: B=n, E=n; P(B=n,E=n) is rare P(A)=? P(C| B=n,E=n, A=a 0, D=d 0 ) => c 1 P(D| B=n,E=n,C=c 1,A=a 0 ) => d 1 P(A| B=n,E=n, D=d 1,C=c 1 ) => a 1 P(C| B=n,E=n, A=a 1, D=d 1 ) => c 2.. discard P(C| B=n,E=n, A=a t-1, D=d t-1 ) => c t. collect.

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Belief Propagation, Junction Trees, and Factor Graphs

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