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Bayesian Networks Unit 6 Exact Inference in Bayesian Networks Wang, Yuan-Kai, 王元凱 ykwang@mails.fju.edu.tw http://www.ykwang.tw Department of Electrical Engineering, Fu Jen Univ. 輔仁大學電機工程系 2006~2011 Reference this document as: Wang, Yuan-Kai, “Exact Inference in Bayesian Networks," Lecture Notes of Wang, Yuan-Kai, Fu Jen University, Taiwan, 2011.Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 2 Goal of This Unit • Learn to efficiently compute the sum product of the inference formula P( X | E e) P ( X i | Pa ( X i )) hH i 1~ n – Remember: enumeration and multiplication of all P(Xi|Pa(Xi) are not efficient – We will learn other 3 methods for exact inference Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 3 Related Units • Background – Probabilistic graphical model • Next units – Approximate inference algorithms – Probabilistic inference over time Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 4 Self-Study References • Chapter 14, Artificial Intelligence-a modern approach, 2nd, by S. Russel & P. Norvig, Prentice Hall, 2003. • The generalized distributive law, S. M. Aji and R. J. McEliece, IEEE Trans. On Information Theory, vol. 46, no. 2, 2000. • Inference in Bayesian networks, B. D’Ambrosio, AI Magazine, 1999. • Probabilistic Inference in graphical models, M. I. Jordan & Y. Weiss. Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 5 Structure of Related Lecture Notes Problem Structure Data Learning PGM B E Representation Learning A Unit 5 : BN Units 16~ : MLE, EM Unit 9 : Hybrid BN J M Units 10~15: Naïve Bayes, MRF, HMM, DBN, Kalman filter P(B) Parameter P(E) Learning P(A|B,E) P(J|A) Query Inference P(M|A) Unit 6: Exact inference Unit 7: Approximate inference Unit 8: Temporal inference Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 6 Contents 1. Basics of Graph ……………………………… 11 2. Sum-Product and Generalized Distributive Law …………………………………………..... 20 3. Variable Elimination ........................................ 29 4. Belief Propagation ....……............................... 96 5. Junction Tree ……………...……………........ 157 6. Summary .......................................................... 212 7. Implementation ……………………………… 214 8. Reference .......................................................... 215 Fu Jen University Fu Jen University Department of Electrical Engineering Department of Electronic Engineering Wang, Yuan-Kai Copyright Yuan-Kai Wang Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 7 Four Steps of Inference P(X|e) • Step 1: Bayesian theorem P ( X , E e) P ( X | E e) P ( X , E e) P ( E e) • Step 2: Marginalization P( X , E e, H h) hH • Step 3: Conditional independence P( X i | Pa ( X i )) hH i 1~ n • Step 4: Sum-Product computation – Exact inference – Approximate inference Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 8 Five Types of Queries in Inference • For a probabilistic graphical model G • Given a set of evidence E=e • Query the PGM with – P(e) : Likelihood query – arg max P(e) : Maximum likelihood query – P(X|e) : Posterior belief query – arg maxx P(X=x|e) : (Single query variable) Maximum a posterior (MAP) query – arg maxx …x P(X1=x1, …, Xk=xk|e) : 1 k Most probable explanation (MPE) query Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 9 Brute Force Enumeration • We can compute in O(KN) time, where K=|Xi| B E A J M • By using BN, we can represent joint distribution in O(N) space Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 10 Expression Tree of Enumeration : Repeated Computations • P(b|j,m)= EAP(b)P(E)P(A|b,E)P(j|A)P(m|A) E=e + E= e + + A=a * A=a * * * A= a * * * A= a * * * * * * * P(a|b,e) P(a|b,e) P(m|a) P(e) P(b) * P(a|b,e)P(m|a) * P(e) P(b) P(a|b,e) P(j|a) P(m|a) P(e) P(b) P(j|a) P(m|a) P(e) P(b) P(j|a) P(j|a) Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 11 1. Basics of Graph • Polytree • Multiply connected networks • Clique • Markov network • Chordal graph • Induced width Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 12 Two Kinds of PGMs • There are two kinds of probabilistic graphical models (PGMs) – Singly connected network • Polytree – Multiply connected network Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 13 Singly Connected Networks (Polytree) • Any two nodes are Burglary Earthquake connected by at most one undirected path Alarm • Theorem John Calls Mary Calls • Inference in a polytree is linear in the node size A H of the network B C • This assumes tabular CPT representation D E F G Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 14 Multiply Connected Networks • At least two nodes are connected by more than one undirected path Cloudy Sprinkler Rain Wet Grass Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 15 Clique (1/2) • A clique is a subgraph of an undirected graph that is complete and maximal – Complete: • Fully connected • Every node connects to every other nodes – Maximal: Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 16 Clique (2/2) • Identify cliques A EGH CEG B C G DEF ACE D E H F ABD ADE Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 17 Markov Network (1/2) • An undirected graph with – Hyper-nodes (multi-vertex nodes) – Hyper-edges (multi-vertex edges) EGH CEG DEF ACE ABD ADE Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 18 Markov Network (2/2) • Every hyper-edge e=(x1…xk) has a potential function fe(x1…xk) • The probability distribution is P ( X 1 ,..., X n ) Z f e ( x e1 ,..., x ek ) e E Z 1 / ... f e ( x e1 ,..., x ek ) x1 xn e E EGH CEG P ( EGH , CEG ) Z f e ( E , G, H , C ) eE Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 19 Chordal Graphs • Elimination ordering undirected chordal graph V S V S T L T L A B A B X D X D Graph: • Maximal cliques are factors in elimination • Factors in elimination are cliques in the graph • Complexity is exponential in size of the largest clique in graph Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 20 2. Sum-Product and Generalized Distributive Law P ( X | E e) P ( X i | Pa ( X i )) hH i 1~ n We obtain the formula because two rules in probability theory Sum Rule : P( x) P( x, y ) y Product Rule : P( x, y ) P( x | y ) P( y ) Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 21 The Sum-Product with Generalized Distributive Law P ( X | E e) P ( X i | Pa ( X i )) hH i 1~ n P ( X i | Pa ( X i )) Xk X 1 i 1~ k P ( X 1 | Pa ( X 1 )) P ( X k | Pa ( X k )) Xk X1 P( X k | Pa ( X k )) P( X t | X k , ) Xk X k 1 P( X X1 1 | Pa ( X 1 )) P( X u | X 1 , ) Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 22 Distributive Law for Sum-Product (1/3) • ax1 ax2 a ( x1 x2 ) ax i i a xi i • x x i j i j ( x)( x) i i j j P( x | x ) i i h i P ( xi , x h ) P ( xh ) P ( xi , xh ) P ( xh ) • P ( x i ) P ( x j ) P ( x i ) P ( x j ) i Variable i i j i j is eliminated P( x | x ) P( x i j i h j | xk ) ( i )( P ( x i | xh ) j P ( x j | xk ) ) P ( xh ) P ( xk ) f1 ( xh ) f 2 ( xk ) Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 23 Distributive Law for Sum-Product (2/3) • P ( x i | xh ) P ( x j | x k ) i j ( i P ( x i | xh ) )( j ) P ( x j | xk ) f1 ( xh ) f 2 ( xk ) • P( x i | xk ) P( x j | xi ) P( x | x )( P( x | x ) ) i k j i i j i j P( x | x ) f ( x )) f ( x ) ( i k i k i • P(b | j , m) P(b) P(e)P(a | b, e) P( j | a) P(m | a) e a P (b) P(e) P (a | b, e) P ( j | a ) P (m | a ) e a Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 24 Distributive Law for Sum-Product (3/3) ab + ac = a(b+c) Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 25 Distributive Law for Max-Product • max(ax1 , ax2 ) a max( x1 , x2 ) max axi a max xi i i • max max xi x j max xi max x j i j i j • max max P ( x i ) P ( x j ) max P ( x i ) max P ( x j ) i j i j max max P( x i | xk ) P( x j | xk ) i j max P( x i | xk ) max P( x j | xk ) i j • arg max P ( xi ) i Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 26 Generalized Distributive Law (1/2) Aji and McEliece, 2000 Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 27 Generalized Distributive Law (2/2) Aji and McEliece, 2000 •a+0=0+a=a •a*1=1*a=a •a*b+a*c=a*(b+c) •max(a,0)=max(0+a)=a •a*1=1*a=a •max(a*b, a*c) =a*max(b, c) Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 28 Marginal to MAP : MAX Product Likelihood & Posterior Queries x1 x2 x3 x4 x5 Maximum Likelihood Query & MAP Query Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 29 3. Variable Elimination • Variable elimination improves the enumeration algorithm by – Eliminating repeated calculations • Carry out summations right-to-left –Bottom-up in the evaluation tree • Storing intermediate results (factors) to avoid re-computation – Dropping irrelevant variables Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 30 Basic Idea • Write query in the form P ( X n , e ) P ( xi | pa i ) xk x3 x2 i • Iteratively –Move all irrelevant terms (constants) outside the innermost summation (i aibc) = (bc (i ai )) –Perform innermost sum, getting a new term: factors –Insert the new term into the product Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 31 An Example without Evidence (1/2) P(C) Cloudy 0.5 C P(R|C) T 0.8 C P(S|C) F 0.2 T 0.1 F 0.5 Sprinkler Rain S R P(W|S,R) T T 0.99 T F 0.90 F T 0.90 WetGrass F F 0.00 P ( w) P ( w | r , s ) P ( r | c ) P ( s | c ) P (c ) r , s ,c P ( w | r , s ) P ( r | c ) P ( s | c ) P (c ) r ,s c P ( w | r , s ) f1 ( r , s ) f1 ( r , s ) r ,s Factor Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 32 An Example without Evidence (2/2) R S C P(R|C) P(S|C) P(C) P(R|C) P(S|C) P(C) T T T T T F T F T T F F F T T F T F F F T F F F R S f1(R,S) = ∑c P(R|S) P(S|C) P(C) Factor f1(r,s) T T A factor may be T F • A function F T • A value F F Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 33 An Example with Evidence (1/2) Factors Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 34 An Example with Evidence (2/2) P(E) Burglary Earthquake • fM(a) = <0.7,0.1> P(B) 0.002 B E P(A|B,E) T T 0.95 • fJ(a) = <0.9,0.05> 0.001 Alarm T F F T 0.95 0.29 • fA(a,b,e) A P(J|A) F F 0.001 John Calls T 0.90 Mary Calls A P(M|A) • fÃJM(b,e) F 0.05 T F 0.70 0.01 J M A B E fM(a) PJ(a) fA(a,b,e) fJM (a,b,e) fÃJM (b,e) T T T T T 0.7 0.9 0.95 0.7*0.9*0.95 T T T T F 0.7 0.9 0.95 0.7*0.9*0.95 T T T F T 0.7 0.9 0.29 0.7*0.9*0.29 T T T F F 0.7 0.9 0.001 0.7*0.9*0.01 T T F T T 0.1 0.05 0.05 0.1*0.05*0.05 T T F T F 0.1 0.05 0.05 0.1*0.05*0.05 T T F F T 0.1 0.05 0.71 0.1*0.05*0.71 T T F F F 0.1 0.05 0.95 0.1*0.05*0.95 Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 35 Basic Operations • Summing out a variable from a product of factors – Move any irrelevant terms (constants) outside the innermost summation – Add up submatrices in pointwise product of remaining factors Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 36 Variable Elimination Algorithm Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 37 Irrelevant Variables (1/2) • Consider the query P(JohnCalls|Burglary = true) – P(J|b)= P(b) eP(e) aP(a|b,e)P(J|a) mP(m|a) – Sum over m is identically 1 mP(m|a) = 1 – M is irrelevant to the query Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 38 Irrelevant Variables (2/2) • Theorem 1: P(X|E) Y is irrelevant if YAncestors({X}E) • In the example P(J|b) – X =JohnCalls, E={Burglary} – Ancestors({X} E) = {Alarm,Earthquake} – so MaryCalls is irrelevant Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 39 Complexity • Time and space cost of variable elimination are O(dkn) – n: No. of random variables – d: no. of discrete values – k: no. of parent nodes k is critical for • Polytrees : k is small, Linear complexity – If k=1, O(dn) • Multiply connected networks : – O(dkn), k is large – Can reduce 3SAT to variable elimination • NP-hard – Equivalent to counting 3SAT models • #P-complete, i.e. strictly harder than NP-complete problems Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 40 Pros and Cons • Variable elimination is simple and efficient for single query P(Xi | e) • But it is less efficient if all the variables are computed: P(X1 | e), …, P(Xk | e) – In a polytree network, one would need to issue O(n) queries costing O(n) each: O(n2) • Junction tree algorithm extends variable elimination that compute posterior probabilities for all nodes simultaneously Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 41 3.1 An Example • The Asia network Visit to Smoking Asia Tuberculosis Lung Cancer Abnormality Bronchitis in Chest X-Ray Dyspnea Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 42 V S • We want to inference P(d) • Need to eliminate: v,s,x,t,l,a,b T L A B Initial factors X D P (v, s , t , l , a , b, x, d ) P ( v ) P ( s ) P (t | v ) P (l | s ) P (b | s ) P ( a | t , l ) P ( x | a ) P ( d | a , b ) “Brute force approach” P (d) P (v, s, t, l, a,b, x, d) x b a l t s v T Complexity is exponential O(N ) • N : size of the graph, number of variables • K : number of states for each variable Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 43 V S • We want to inference P(d) • Need to eliminate : v,s,x,t,l,a,b T L A B Initial factors X D P (v )P ( s )P (t | v )P (l | s )P (b | s )P (a | t , l )P ( x | a )P (d | a , b ) Eliminate: v Compute: fv (t ) P (v )P (t |v ) v fv (t )P ( s )P (l | s )P (b | s )P (a | t , l )P ( x | a )P (d | a , b ) t fv(t) Note: fv(t) = P(t) T 0.70 In general, result of elimination is F 0.01 not necessarily a probability term Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 44 V S • We want to inference P(d) • Need to eliminate : s,x,t,l,a,b T L A B • Initial factors X D P (v )P ( s )P (t | v )P (l | s )P (b | s )P (a | t , l )P ( x | a )P (d | a , b ) fv (t )P ( s )P (l | s )P (b | s )P (a | t , l )P ( x | a )P (d | a , b ) Eliminate: s Compute: fs (b , l ) P (s )P (b | s )P (l | s ) s fv (t )fs (b , l )P (a | t , l )P ( x | a )P (d | a , b ) b l fs(b,l) T T 0.95 •Summing on s results in fs(b,l) T F F 0.95 T 0.29 •A factor with two arguments F F 0.001 •Result of elimination may be a function of several variables Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 45 V S • We want to inference P(d) • Need to eliminate : x,t,l,a,b T L A B • Initial factors X D P (v )P ( s )P (t | v )P (l | s )P (b | s )P (a | t , l )P ( x | a )P (d | a , b ) fv (t )P ( s )P (l | s )P (b | s )P (a | t , l )P ( x | a )P (d | a , b ) fv (t )fs (b , l )P (a | t , l )P ( x | a )P (d | a , b ) Eliminate: x Compute: fx (a ) P (x | a ) x fv (t )fs (b , l )fx (a )P (a | t , l )P (d | a , b ) Note: fx(a) = 1 for all values of a !! Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 46 V S • We want to inference P(d) • Need to eliminate : t,l,a,b T L A B • Initial factors X D P (v )P ( s )P (t | v )P (l | s )P (b | s )P (a | t , l )P ( x | a )P (d | a , b ) fv (t )P ( s )P (l | s )P (b | s )P (a | t , l )P ( x | a )P (d | a , b ) fv (t )fs (b , l )P (a | t , l )P ( x | a )P (d | a , b ) fv (t )fs (b , l )fx (a )P (a | t , l )P (d | a , b ) Eliminate: t Compute: ft (a , l ) fv (t )P (a |t , l ) t fs (b , l )fx (a )ft (a , l )P (d | a , b ) Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 47 V S • We want to inference P(d) • Need to eliminate : l,a,b T L A B • Initial factors X D P (v )P ( s )P (t | v )P (l | s )P (b | s )P (a | t , l )P ( x | a )P (d | a , b ) fv (t )P ( s )P (l | s )P (b | s )P (a | t , l )P ( x | a )P (d | a , b ) fv (t )fs (b , l )P (a | t , l )P ( x | a )P (d | a , b ) fv (t )fs (b , l )fx (a )P (a | t , l )P (d | a , b ) fs (b , l )fx (a )ft (a , l )P (d | a , b ) Eliminate: l Compute: fl (a , b ) fs (b , l )ft (a , l ) l fl (a , b )fx (a )P (d | a , b ) Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 48 V S • We want to inference P(d) T L • Need to eliminate : b A B • Initial factors X D P (v )P ( s )P (t | v )P (l | s )P (b | s )P (a | t , l )P ( x | a )P (d | a , b ) fv (t )P ( s )P (l | s )P (b | s )P (a | t , l )P ( x | a )P (d | a , b ) fv (t )fs (b , l )P (a | t , l )P ( x | a )P (d | a , b ) fv (t )fs (b , l )fx (a )P (a | t , l )P (d | a , b ) fs (b , l )fx (a )ft (a , l )P (d | a , b ) fl (a , b )fx (a )P (d | a , b ) fa (b , d ) fb (d ) Eliminate: a,b Compute: fa (b , d ) fl (a , b )fx (a ) p (d | a , b ) a fb (d ) fa (b , d ) b Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 49 V S • Different elimination ordering • Need to eliminate : a,b,x,t,v,s,l T L • Initial factors A B P (v)P (s)P (t | v)P (l | s)P (b | s)P (a | t, l)P ( x | a)P (d | a,b) X D Intermediate factors: In previous order g a (l , t , d , b , x , s , v ) Both f v (v, s , x, t , l , a , b ) g b (l , t , d , x , s , v ) need f s ( s , x, t , l , a , b ) g x (l , t , d , s , v ) n=7 f x ( x, t , l , a , b ) g t (l , d , s , v ) steps f t (t , l , a, b) g v (l , d , s ) f l (l , a, b) g s (l , d ) But each step has f a ( a, b) different g l (d ) computation size f b (d ) Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 50 Short Summary • Variable elimination is a sequence of rewriting operations • Computation depends on – Number of variables n • Each elimination step reduces one variable • So we need n elimination steps – Size of factors • Effected by order of elimination • Discussed in sub-section 3.2 Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 51 V S Dealing with Evidence(1/7) T L A B • How do we deal with evidence? X D • Suppose get evidence V = t, S = f, D = t • We want to compute P(L, V = t, S = f, D = t) Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 52 V S Dealing with Evidence(2/7) T L A B • We start by writing the factors: X D P (v )P ( s )P (t | v )P (l | s )P (b | s )P (a | t , l )P ( x | a )P (d | a , b ) • Since we know that V = t, we don’t need to eliminate V • Instead, we can replace the factors P(V) and P(T|V) with fP (V ) P (V t ) fp (T |V ) ( ) P ( |V t ) T T • These “select” the appropriate parts of the original factors given the evidence • Note that fp(V) is a constant, and thus does not appear in elimination of other variables Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 53 V S Dealing with Evidence(3/7) T L • Given evidence V = t, S = f, D = t A B • Compute P(L, V = t, S = f, D = t ) X D • Initial factors, after setting evidence: fP (v )fP ( s )fP (t |v ) (t )fP (l |s ) (l )fP ( b|s ) (b )P (a | t , l )P ( x | a )fP (d |a ,b ) (a , b ) Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 54 V S Dealing with Evidence(4/7) T L A B • Given evidence V = t, S = f, D = t • Compute P(L, V = t, S = f, D = t ) X D • Initial factors, after setting evidence: fP (v )fP ( s )fP (t |v ) (t )fP (l |s ) (l )fP ( b|s ) (b )P (a | t , l )P ( x | a )fP (d |a ,b ) (a , b ) • Eliminating x, we get fP (v )fP ( s )fP (t |v ) (t )fP (l |s ) (l )fP ( b|s ) (b )P (a | t , l )fx (a )fP (d |a ,b ) (a , b ) Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 55 V S Dealing with Evidence(5/7) T L • Given evidence V = t, S = f, D = t A B • Compute P(L, V = t, S = f, D = t ) • Initial factors, after setting evidence: X D fP (v )fP ( s )fP (t |v ) (t )fP (l |s ) (l )fP ( b|s ) (b )P (a | t , l )P ( x | a )fP (d |a ,b ) (a , b ) • Eliminating x, we get fP (v )fP ( s )fP (t |v ) (t )fP (l |s ) (l )fP ( b|s ) (b )P (a | t , l )fx (a )fP (d |a ,b ) (a , b ) • Eliminating t, we get fP (v )fP ( s )fP (l |s ) (l )fP ( b|s ) (b )ft (a , l )fx (a )fP (d |a ,b ) (a , b ) Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 56 V S Dealing with Evidence(6/7) T L • Given evidence V = t, S = f, D = t A B • Compute P(L, V = t, S = f, D = t ) • Initial factors, after setting evidence: X D fP (v )fP ( s )fP (t |v ) (t )fP (l |s ) (l )fP ( b|s ) (b )P (a | t , l )P ( x | a )fP (d |a ,b ) (a , b ) • Eliminating x, we get fP (v )fP ( s )fP (t |v ) (t )fP (l |s ) (l )fP ( b|s ) (b )P (a | t , l )fx (a )fP (d |a ,b ) (a , b ) • Eliminating t, we get fP (v )fP ( s )fP (l |s ) (l )fP ( b|s ) (b )ft (a , l )fx (a )fP (d |a ,b ) (a , b ) • Eliminating a, we get fP (v )fP ( s )fP (l |s ) (l )fP ( b|s ) (b )fa (b , l ) Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 57 V S Dealing with Evidence(7/7) T L • Given evidence V = t, S = f, D = t A B • Compute P(L, V = t, S = f, D = t ) • Initial factors, after setting evidence: X D fP (v )fP ( s )fP (t |v ) (t )fP (l |s ) (l )fP ( b|s ) (b )P (a | t , l )P ( x | a )fP (d |a ,b ) (a , b ) • Eliminating x, we get fP (v )fP ( s )fP (t |v ) (t )fP (l |s ) (l )fP ( b|s ) (b )P (a | t , l )fx (a )fP (d |a ,b ) (a , b ) • Eliminating t, we get fP (v )fP ( s )fP (l |s ) (l )fP ( b|s ) (b )ft (a , l )fx (a )fP (d |a ,b ) (a , b ) • Eliminating a, we get fP (v )fP ( s )fP (l |s ) (l )fP ( b|s ) (b )fa (b , l ) • Eliminating b, we get fP (v )fP ( s )fP (l |s ) (l )fb (l ) Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 58 Complexity (1/2) • Suppose in one elimination step we compute fx ( y 1 , , y k ) f x (x , y , , y k ) x 1 m f x ( x , y , , y k ) fi ( x , y , y 1 1,1, 1,li ) i 1 This requires |X| : No. of discrete values of X • m X Yi multiplications i – For each value for x, y1, …, yk, we do m multiplications • X Yi additions i – For each value of y1, …, yk , we do |X| additions Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 59 Complexity (2/2) • One elimination step requires – m X Yi multiplications i – X Yi additions i – O( X Yi ), m is a constant (neglected) i – Or O(d k) if • |X|=|Yi|=d, • k: no. of parent nodes • Time and space cost are O(dkn) Complexity is – n: No. of random variables exponential in number – d: no. of discrete values of variables k – k: no. of parent nodes Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 60 3.2 Order of Elimination • How to select “good” elimination orderings in order to reduce complexity 1. Start by understanding variable elimination via the graph we are working with 2. Then reduce the problem of finding good ordering to graph-theoretic operation that is well-understood Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 61 Undirected Graph Conversion (1/2) • At each stage of the variable elimination, • We have an algebraic term that we need to evaluate • This term is of the form P ( x 1 , , x k ) fi ( Z i ) y1 yn i where Zi are sets of variables Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 62 Undirected Graph Conversion (2/2) • Plot a graph where – If X,Y are arguments of some factor • That is, if X,Y are in some Zi – There are undirected edges X--Y Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 63 Example • Consider the “Asia” example • The initial factors are P (v )P ( s )P (t | v )P (l | s )P (b | s )P (a | t , l )P ( x | a )P (d | a , b ) • The undirected graph is V S V S T L T L A B A B X D X D • In the first step this graph is just the moralized graph Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 64 Variable Elimination Change of Graph P (v )P ( s )P (t | v )P (l | s )P (b | s )P (a | t , l )P ( x | a )P (d | a , b ) • Now we eliminate t, getting P (v )P ( s )P (l | s )P (b | s )P ( x | a )P (d | a , b )ft (v , a , l ) • The corresponding change in the graph is V S V S T L T L Nodes V,L,A become A B A B a clique X D X D Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 65 Example (1/6) • Want to compute P(L,V=t,S=f,D=t) V S T L A B • Moralizing V S X D T L A B X D Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 66 Example (2/6) • Want to compute P(L,V=t,S=f,D=t) V S T L • Moralizing A B • Setting evidence X D V S T L A B X D Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 67 Example (3/6) • Want to compute P(L,V=t,S=f,D=t) V S T L • Moralizing A B • Setting evidence • Eliminating x X D V S – New factor fx(A) T L A B X D Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 68 Example (4/6) • Want to compute P(L,V=t,S=f,D=t) V S T L • Moralizing A B • Setting evidence X D • Eliminating x Eliminating a V S • – New factor fa(b,t,l) T L A B A clique in reduced undirected graph X D Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 69 Example (5/6) • Want to compute P(L,V=t,S=f,D=t) V S • Moralizing T L • Setting evidence A B • Eliminating x X D • Eliminating a V S • Eliminating b T L – New factor fb(t,l) A B A clique in reduced X D undirected graph Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 70 Example (6/6) • Want to compute P(L,V=t,S=f,D=t) V S T L • Moralizing A B • Setting evidence X D • Eliminating x V S • Eliminating a T L • Eliminating b Eliminating t A B • – New factor ft(l) X D Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 71 Elimination and Clique (1/2) • We can eliminate a variable x by 1. For all Y,Z, s.t., Y--X, Z--X • add an edge Y--Z 2. Remove X and all adjacent edges to it • This procedures create a clique that contains all the neighbors of X • After step 1 we have a clique that corresponds to the intermediate factor (before marginalization) • The cost of the step is exponential in the size of this clique : dk in O(ndk) Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 72 Elimination and Clique (2/2) • The process of eliminating nodes from an undirected graph gives us a clue to the complexity of inference • To see this, we will examine the graph that contains all of the edges we added during the elimination • The resulting graph is always chordal Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 73 V S Example (1/7) T L • Want to compute P(L) A B X D • Moralizing V S T L A B X D Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 74 V S Example (2/7) T L • Want to compute P(L) A B X D • Moralizing • Eliminating v V S – Multiply to get f’v(v,t) – Result fv(t) T L A B X D Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 75 V S Example (3/7) T L • Want to compute P(L) A B X D • Moralizing • Eliminating v V S • Eliminating x T L –Multiply to get f’x(a,x) –Result fx(a) A B X D Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 76 V S Example (4/7) T L • Want to compute P(L) A B X D • Moralizing • Eliminating v V S • Eliminating x T L • Eliminating s –Multiply to get f’s(l,b,s) A B –Result fs(l,b) X D Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 77 V S Example (5/7) T L • Want to compute P(D) A B X D • Moralizing • Eliminating v • Eliminating x V S • Eliminating s T L • Eliminating t A B –Multiply to get f’t(a,l,t) X D –Result ft(a,l) Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 78 V S Example (6/7) T L • Want to compute P(D) A B X D • Moralizing • Eliminating v V S • Eliminating x T L • Eliminating s • Eliminating t A B • Eliminating l X D –Multiply to get f’l(a,b,l) –Result fl(a,b) Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 79 V S Example (7/7) T L • Want to compute P(D) A B X D • Moralizing • Eliminating v • Eliminating x V S • Eliminating s T L • Eliminating t A B • Eliminating l X D • Eliminating a, b –Multiply to get f’a(a,b,d) –Result f(d) Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 80 Induced Graphs V S • The resulting graph are induced T L graphs (for this particular ordering) A B X D • Main property: – Every maximal clique in the induced graph corresponds to an intermediate factor in the computation – Every factor stored during the process is a subset of some maximal clique in the graph • These facts are true for any variable elimination ordering on any network Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 81 Induced Width (Treewidth) • The size of the largest clique k in the induced graph is – An indicator for the complexity of variable elimination • w=k-1 is called – Induced width (treewidth) of a graph – According to the specified ordering • Finding a good ordering for a graph is equivalent to finding the minimal induced width of the graph Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 82 Treewidth Low treewidth High tree width Chains N=nxn grid W=1 Trees (no loops) W = O(n) = O(p N) MINVOLSET INTUBATION KINKEDTUBE PULMEMBOLUS VENTMACH DISCONNECT PAP SHUNT VENTLUNG VENITUBE MINOVL PVSAT VENTALV ARTCO2 PRESS Loopy graphs Arnborg85 TPR SAO2 EXPCO2 INSUFFANESTH HYPOVOLEMIA LVFAILURE CATECHOL LVEDVOLUME STROEVOLUME ERRBLOWOUTPUT HISTORY HRERRCAUTER CVP PCWP CO HREKGHRSAT HRBP BP W = #parents W = NP-hard to find Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 83 Complexity • Time and space cost of variable elimination are O(dkn) – n: No. of random variables – d: no. of discrete values – k: no. of parent nodes = treewidth + 1 (W+1) • Polytrees : k is small, Linear – If k=1, O(dn) • Multiply connected networks : – O(dkn), k is large – Can reduce 3SAT to variable elimination • NP-hard – Equivalent to counting 3SAT models • #P-complete, i.e. strictly harder than NP-complete problems Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 84 Elimination on Trees (1/3) • Suppose we have a tree that – A network where each variable has at most one parent • Then all the factors involve at most two variables: Treewidth=1 • The moralized graph is also a tree A A B C B C D E D E F G F G Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 85 Elimination on Trees (2/3) • We can maintain the tree structure by eliminating extreme variables in the tree A A B C B C D E D E A F G F G B C D E F G Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 86 Elimination on Trees (3/3) • Formally, for any tree, there is an elimination ordering with treewidth = 1 Theorem • Inference on trees is linear in number of variables : O(dn) Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 87 Exercise: Variable Elimination p(smart)=.8 p(study)=.6 Query: What is the probability smart study that a student studied, given that they pass the exam? p(fair)=.9 prepared fair p(prep|…) smart smart pass study .9 .7 smart smart study .5 .1 p(pass|…) prep prep prep prep fair .9 .7 .7 .2 fair .1 .1 .1 .1 Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 88 Variable Elimination Algorithm • Let X1,…, Xm be an ordering on the non-query variables ... P ( X | Parents ( X )) j j X 1 X 2 X m j • For i = m, …, 1 – Leave in the summation for Xi only factors mentioning Xi – Multiply the factors, getting a factor that contains a number for each value of the variables mentioned, including Xi – Sum out Xi, getting a factor f that contains a number for each value of the variables mentioned, not including Xi – Replace the multiplied factor in the summation Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 89 3.3 General Graphs • If the graph is not a polytree – More general networks – Usually loopy networks • Can we inference loopy networks by variable elimination? – If network has a cycle, the treewidth for any ordering is greater than 1 – Its complexity is high, – VE becomes a not practical algorithm Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN A p. 90 B C Example (1/2) D E • Eliminating A, B, C, D, E,…. F G • Resulting graph is chordal with treewidth 2 H A A A A B C B C B C B C D E D E D E D E F G F G F G F G H H H H Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN A p. 91 B C Example (2/2) D E • Eliminating H,G, E, C, F, D, E, A F G • Resulting graph is chordal with treewidth 3 H A A A A B C B C B C B C D E D E D E D E F G F G F G F G H H H H Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 92 Find Good Elimination Order in General Graph Theorem: • Finding an ordering that minimizes the treewidth is NP-Hard However, • There are reasonable heuristic for finding “relatively” good ordering • There are provable approximations to the best treewidth • If the graph has a small treewidth, there are algorithms that find it in polynomial time Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 93 Heuristics for Finding an Elimination Order • Since elimination order is NP-hard to optimize, • It is common to apply greedy search techniques: Kjaerulff90 • At each iteration, eliminate the node that would result in the smallest – Number of fill-in edges [min-fill] – Resulting clique weight [min-weight] (Weight of clique = product of number of states per node in clique) • There are some approximation algorithms Amir01 Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 94 Factorization in Loopy Networks Probabilistic models with no loop are tractable Factorizable a b Pa, x P(b, x) P(c, x) P(d, x) a b c d c d P (a, x) P (b, x) P (c, x) P (d, x) a b c d Probabilistic models with loop are not tractable a Not Factorizable b c Pa, b, c, d, x a b c d d Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 95 Short Summary • Variable elimination – Actual computation is done in elimination step – Computation depends on order of elimination – Very sensitive to topology – Space = time • Complexity – Polytrees: Linear time – General graphs: NP-hard Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 96 4. Belief Propagation • Also called – Message passing – Pearl’s algorithm • Subsections – 4.1 Message passing in simple chains – 4.2 Message passing in trees – 4.3 BP Algorithm – 4.4 Message passing in general graphs Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 97 What’s Wrong with VarElim • Often we want to query all hidden nodes • Variable elimination takes O(N2dk) time to compute P(Xi|e) for all (hidden) nodes Xi • Message passing algorithms that can do this in O(Ndk) time Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 98 Repeated Variable Elimination Leads to Redundant Calculations X1 X2 X3 Y1 Y2 Y3 P ( x1 | y1:3 ) P ( x1 ) P ( y1 | x1 ) P ( x 2 | x1 ) P ( y 2 | x 2 ) P ( x3 | x 2 ) P ( y 3 | x3 ) x2 x3 P ( x 2 | y1:3 ) P ( x 2 | x1 ) P ( y 2 | x 2 ) P ( x1 ) P ( y1 | x1 ) P ( x3 | x 2 ) P ( y 3 | x3 ) x1 x3 P ( x3 | y1:3 ) P ( x3 | x 2 ) P ( y 3 | x3 ) P ( x1 ) P ( y1 | x1 ) P ( x 2 | x1 ) P ( y 2 | x 2 ) x1 x2 O(N2 K2) time to compute all N marginals Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 99 Belief Propagation • Belief propagation (BP) operates by sending beliefs/messages between nearby variables in the graphical model • It works like variable elimination Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 100 4.1 Message Passing in Simple Chains X1 ... Xk ... Xn • Likelihood query (query without evidence) – P(X1), P(Xn), P(Xk) – P(Xj , Xk) • Posterior query (query with evidence) – P(X1|Xn), P(Xn|X1), – P(Xk|X1), P(Xk|Xn), – P(X1|Xk), P(Xn|Xk), – P(Xk|Xj) • Maximum A Posterior (MAP) query – arg max P(Xk|Xj) Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 101 Sum-Product of the Simple Chain (1/2) X1 ... Xk ... Xn P( X k ) P( X X 1 X k 1 , X k 1 X n 1 , , X k , , X n ) P ( X 1 , , X k , , X n ) X1 X k 1 X k 1 Xn P ( X i | Pa ( X i )) X1 X k 1 X k 1 Xn Xi P ( X n | X n 1 ) P ( X k | X k 1 ) P ( X 2 | X 1 ) P ( X 1 ) X1 X k 1 X k 1 Xn P ( X 1 ) P ( X 2 | X 1 ) P ( X k 1 | X k 2 ) P ( X k | X k 1 ) X1 X2 X k 1 P( X X k 1 k 1 | X k ) P ( X n | X n 1 ) Xn Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 102 Sum-Product of the Simple Chain (2/2) X1 ... Xk ... Xn P( X k | X j ) P ( X 1 , , X n ) { X i |1 i n , i j , k } P( X { X i |1 i n , i j , k } X i i | Pa ( X i )) P( X { X i |1 i n , i j , k } n | X n 1 ) P ( X k | X k 1 ) P ( X 2 | X 1 ) P ( X 1 ) Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 103 4.1.1 Likelihood Query • P(Xn) or P(xn) : Forward passing X1 X2 X3 ... Xn • P(X1) or P(x1) : Backward passing X1 X2 X3 ... Xn • P(Xk) or P(xk) : Forward-Backward passing X1 X2 ... Xk ... Xn Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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Bayesian Networks Unit - Exact Inference in BN p. 104 Forward Passing (1/6) A B C D E • P(e) P ( e ) P ( a ) P (b | a ) P ( c | b ) P ( d | c ) P ( e | d ) d c b a P ( e | d ) P ( d | c ) P ( c | b ) P ( a ) P (b | a ) d c b a Fu Jen University Department of Electrical Engineering Wang, Yuan-Kai Copyright
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