Junction tree propagation - BNDG 4-4.6

Transcription

1 Junction tree propagation - BNDG 4-4. Finn V. Jensen and Thomas D. Nielsen Junction tree propagation p. 1/2

2 Exact Inference Message Passing in Join Trees More sophisticated inference technique; used in most implemented Bayesian Network systems (e.g. Hugin). Overview: Given: Bayesian network for P(V) Preprocessing Construct a new internal representation for P(V) called a junction tree Inference/Updating Retrieve P(A E = e) for single A V Junction tree propagation p. 2/2

3 Calculate a marginal P(A 4 ) = X X X X X A 1 A 2 A 3 A A φ 1 (A 1 )φ 2 (A 2, A 1 )φ 3 (A 3, A 1 )φ 4 (A 4, A 2 )φ (A, A 2, A 3 )φ (A, A 3 ) Junction tree propagation p. 3/2

4 Calculate a marginal P(A 4 ) = X X X X X A 1 A 2 A 3 A A φ 1 (A 1 )φ 2 (A 2, A 1 )φ 3 (A 3, A 1 )φ 4 (A 4, A 2 )φ (A, A 2, A 3 )φ (A, A 3 ) = X A 1 φ 1 (A 1 ) X A 2 φ 2 (A 2, A 1 )φ 4 (A 4, A 2 ) X A 3 φ 3 (A 3, A 1 ) X A φ (A, A 2, A 3 ) X A φ (A, A 3 ) Junction tree propagation p. 3/2

5 Calculate a marginal P(A 4 )= P A 1 φ 1 (A 1 ) P A 2 φ 2 (A 2,A 1 )φ 4 (A 4,A 2 ) P A 3 φ 3 (A 3,A 1 ) P A φ (A,A 2,A 3 ) P A φ (A,A 3 ) Junction tree propagation p. 4/2

6 Calculate a marginal Some of the calculations can be simplified. E.g. X A P(A A 3 ) = 1 P(A 4 )= P A 1 φ 1 (A 1 ) P A 2 φ 2 (A 2,A 1 )φ 4 (A 4,A 2 ) P A 3 φ 3 (A 3,A 1 ) P A φ (A,A 2,A 3 ) P A φ (A,A 3 ) Junction tree propagation p. 4/2

7 Calculate a marginal Assume evidence e = (e,e ); represented as 0/1 potentials. Then: P(A 4,e) = X φ 1 (A 1 ) X φ 2 (A 2, A 1 )φ 4 (A 4, A 2 ) A 1 A 2 X X φ 3 (A 3, A 1 ) φ (A, A 2, A 3 )e A 3 A X φ (A, A 3 )e A and P(A 4 e) = P(A 4,e) P A 4 P(A 4,e) Junction tree propagation p. /2

8 Calculate a marginal Assume evidence e = (e,e ); represented as 0/1 potentials. Then: P(A 4,e) = X φ 1 (A 1 ) X φ 2 (A 2, A 1 )φ 4 (A 4, A 2 ) A 1 A 2 X X φ 3 (A 3, A 1 ) φ (A, A 2, A 3 )e A 3 A X φ (A, A 3 )e A and P(A 4 e) = P(A 4,e) P A 4 P(A 4,e) The process is sufficiently general to handle all evidence scenarios! We look for a general structure in which these calculations can be performed for all variables. Junction tree propagation p. /2

9 Domain graphs The Bayesian network The domain graph The link (A 2, A 3 ) is called a moral link. Junction tree propagation p. /2

10 Domain graphs Moralization in general For all nodes X: Connect pairwise all parents of X with undirected links. Replace all original directed links by undirected ones. The Bayesian network Junction tree propagation p. /2

11 Domain graphs Moralization in general For all nodes X: Connect pairwise all parents of X with undirected links. Replace all original directed links by undirected ones. The Bayesian network Step 1 Junction tree propagation p. /2

12 Domain graphs Moralization in general For all nodes X: Connect pairwise all parents of X with undirected links. Replace all original directed links by undirected ones. The Bayesian network Step 1 Step 2 Junction tree propagation p. /2

13 Eliminating a variable Suppose that when calculating P(A 4 ) we start off by eliminating A 3 : φ (A 1, A 2, A, A ) = X A 3 φ 3 (A 3, A 1 )φ 4 (A 4, A 2 )φ (A, A 2, A 3 ). Graphically: The Bayesian network The domain graph The new domain graph Junction tree propagation p. /2

14 Eliminating a variable Suppose that when calculating P(A 4 ) we start off by eliminating A 3 : φ (A 1, A 2, A, A ) = X A 3 φ 3 (A 3, A 1 )φ 4 (A 4, A 2 )φ (A, A 2, A 3 ). Graphically: The Bayesian network The domain graph The new domain graph Perfect elimination sequences If all variables can be eliminated without introducing fill-in edges, then the elimination sequence is called perfect. An example could be A, A, A 3, A 1, A 2, A 4. Junction tree propagation p. /2

15 Perfect elimination sequences Perfect elimination sequences All perfect elimination sequences produce the same domain set, namely the set of cliques of the domain graph. Clique A set of nodes is complete if all nodes are pairwise linked. A complete set is a clique if it is not a subset of another complete set. Property Any perfect elimination sequence ending with A is optimal w.r.t. calculating P(A). Junction tree propagation p. 9/2

16 Triangulated graphs Definition An undirected graph with a perfect elimination sequence is called a triangulated graph. Example not triangulated triangulated Junction tree propagation p. 10/2

17 Triangulated graphs Definition An undirected graph with a perfect elimination sequence is called a triangulated graph. Example not triangulated triangulated Corollary A graph is triangulated iff all nodes can successively be eliminated without introducing fill-in edges. Junction tree propagation p. 10/2

18 Triangulated graphs Definition An undirected graph with a perfect elimination sequence is called a triangulated graph. Example not triangulated triangulated Corollary A graph is triangulated iff there does not exist a cycle of length 4 that is not cut by a cord. Junction tree propagation p. 10/2

19 Triangulated graphs Definition An undirected graph with a perfect elimination sequence is called a triangulated graph. Example not triangulated triangulated Corollary In a triangulated graph, each variable A has a perfect elimination sequence ending with A. Junction tree propagation p. 10/2

20 Triangulated graphs Definition An undirected graph with a perfect elimination sequence is called a triangulated graph. Example not triangulated triangulated Assumption For now we shall assume that the domain graph is triangulated! Junction tree propagation p. 10/2

21 Exact Inference Join Tree (Junction Tree) Let G be the set of cliques from an undirected graph, and let the cliques of G be organized in a tree T. Then T is a join tree if a variable V that is contained in two nodes C,C also is contained in every node on the (unique) path connecting C and C. Example: Junction tree propagation p. 11/2

22 Exact Inference Join Tree (Junction Tree) Let G be the set of cliques from an undirected graph, and let the cliques of G be organized in a tree T. Then T is a join tree if a variable V that is contained in two nodes C,C also is contained in every node on the (unique) path connecting C and C. Example: (a) A join tree (b) Not a join tree Junction tree propagation p. 11/2

23 Exact Inference Join Tree (Junction Tree) Let G be the set of cliques from an undirected graph, and let the cliques of G be organized in a tree T. Then T is a join tree if a variable V that is contained in two nodes C,C also is contained in every node on the (unique) path connecting C and C. Example: (a) A join tree (b) Not a join tree Theorem: An undirected graph G is triangulated if and only if the cliques of G can be organized in a join tree. Junction tree propagation p. 11/2

24 Join tree construction Define undirected graph (C, E ): From undirected graph to join graph C : Set of cliques in triangulated moral graph E : {(C 1,C 2 ) C 1 C 2 } Junction tree propagation p. 12/2

25 Join tree construction Define undirected graph (C, E ): From undirected graph to join graph C : Set of cliques in triangulated moral graph E : {(C 1,C 2 ) C 1 C 2 } Junction tree propagation p. 12/2

26 Join tree construction Define undirected graph (C, E ): From undirected graph to join graph C : Set of cliques in triangulated moral graph E : {(C 1,C 2 ) C 1 C 2 } Junction tree propagation p. 12/2

27 Join tree construction Define undirected graph (C, E ): From undirected graph to join graph C : Set of cliques in triangulated moral graph E : {(C 1,C 2 ) C 1 C 2 } Junction tree propagation p. 12/2

28 Join tree construction Define undirected graph (C, E ): From undirected graph to join graph C : Set of cliques in triangulated moral graph E : {(C 1,C 2 ) C 1 C 2 } 1,2,3 2,4, 2,3, 3,,,,, Junction tree propagation p. 12/2

29 Join tree construction From join graph to join tree Find maximal spanning tree: for edge e = (C 1,C 2 ) define weight w(e) := C 1 C 2. Let (C, E) be a maximal spanning tree of (C, E ) with respect to edge weights. Junction tree propagation p. 13/2

30 Join tree construction From join graph to join tree Find maximal spanning tree: for edge e = (C 1,C 2 ) define weight w(e) := C 1 C 2. Let (C, E) be a maximal spanning tree of (C, E ) with respect to edge weights. 1,2,3 2,4, 2,3, 3,,,,, Junction tree propagation p. 13/2

31 Join tree construction From join graph to join tree Find maximal spanning tree: for edge e = (C 1,C 2 ) define weight w(e) := C 1 C 2. Let (C, E) be a maximal spanning tree of (C, E ) with respect to edge weights. 1 1,2,3 2,4, ,3, 1 2,,, 3,, 2 Junction tree propagation p. 13/2

32 Join tree construction From join graph to join tree Find maximal spanning tree: for edge e = (C 1,C 2 ) define weight w(e) := C 1 C 2. Let (C, E) be a maximal spanning tree of (C, E ) with respect to edge weights. 1 1,2,3 2,4, ,3, 1 2,,, 3,, 2 Junction tree propagation p. 13/2

33 Join tree construction From join graph to join tree Find maximal spanning tree: for edge e = (C 1,C 2 ) define weight w(e) := C 1 C 2. Let (C, E) be a maximal spanning tree of (C, E ) with respect to edge weights. 1,2,3 2,4, 2,3, 3,,,,, Junction tree propagation p. 13/2

34 Join tree construction Another example Junction tree propagation p. 14/2

35 Join tree construction Another example A 1 A 2 A 3 A 4 A A Junction tree propagation p. 14/2

36 Join tree construction Another example A 1 A 2 A 3 A 4 A A 1 A 1, A 2, A 3 1 A 2, A A 2, A 3, A A 2, A 3, A Junction tree propagation p. 14/2

37 Join tree construction Another example A 1 A 2 A 3 A 4 A A A 1, A 2, A 3 A 1, A 2, A A 2, A 4 2 A 2, A 3, A A 2, A 4 A 2, A 3, A 1 1 A 2, A 3, A A 2, A 3, A Junction tree propagation p. 14/2

38 Junction tree construction Correctness Let (V, ) be a triangulated graph, (C, E ). Every maximal spanning tree (C, E) of (C, E ) is a join tree for V (Jensen,Jensen 1994). If (V, ) is the triangulated moral graph of a Bayesian network, then there exists for every V V a clique C C with {V } pa(v ) C. Junction tree propagation p. 1/2

39 Propagation in junction trees Initialization Let T be a join tree for the domain graph G with potentials Φ. A junction tree for G consists of T with the additions: each potential φ from Φ is assigned to a clique containing dom (φ). each link has a separator attached. each separator contains two mailboxes, one for each direction. Example Junction tree propagation p. 1/2

40 Propagation in junction trees Propagation To calculate P(A 4 ) we find a clique (V ) containing A 4 and send messages to that clique. Example φ 1, φ 2, φ 3 V 4 : A 1, A 2, A 3 ψ 4 ψ 2 = φ S 2 ψ 1 = φ S 1 S 4 : A 2 S 2 : A 2, A 3 S 1 : A 3 φ 4 φ φ V : A 2, A 4 V 2 : A 2, A 3, A V 1 : A 3, A Junction tree propagation p. 1/2

41 Propagation in junction trees Propagation To calculate P(A 4 ) we find a clique (V ) containing A 4 and send messages to that clique. Example φ 1, φ 2, φ 3 V 4 : A 1, A 2, A 3 1 : ψ 2 = φ S 2 1 : ψ 1 = φ S 1 S 4 : A 2 S 2 : A 2, A 3 S 1 : A 3 φ 4 φ φ V : A 2, A 4 V 2 : A 2, A 3, A V 1 : A 3, A Junction tree propagation p. 1/2

42 Propagation in junction trees Propagation To calculate P(A 4 ) we find a clique (V ) containing A 4 and send messages to that clique. Example φ 1, φ 2, φ 3 V 4 : A 1, A 2, A 3 2 : ψ 4 =... 1 : ψ 1 : ψ 1 = φ S 1 2 = φ S 2 S 4 : A 2 S 2 : A 2, A 3 S 1 : A 3 φ 4 φ φ V : A 2, A 4 V 2 : A 2, A 3, A V 1 : A 3, A ψ 4 = P A 1 φ 1 φ 2 PA 3 φ 3 ψ 1 ψ 2 Now evidence has been collected to V 4 and we get P(A 4 ) = P A 2 φ 4 ψ 4. Junction tree propagation p. 1/2

43 Propagation in junction trees Propagation To calculate P(A i ) for any A i we send messages away from V. Example continued φ 1, φ 2, φ 3 V 4 : A 1, A 2, A 3 2 : ψ 4 =... 1 : ψ 1 : ψ 1 = φ S 1 2 = φ S 2 S 4 : A 2 S 2 : A 2, A 3 S 1 : A 3 φ 4 φ φ V : A 2, A 4 V 2 : A 2, A 3, A V 1 : A 3, A ψ 4 = P A 1 φ 1 φ 2 PA 3 φ 3 ψ 1 ψ 2 Junction tree propagation p. 1/2

44 Propagation in junction trees Propagation To calculate P(A i ) for any A i we send messages away from V. Example continued φ 1, φ 2, φ 3 V 4 : A 1, A 2, A 3 2 : ψ 4 =... 1 : ψ 1 : ψ 1 = φ S 1 2 = φ S 2 3 : ψ 4 = φ A 2 4 S 4 : A 2 S 2 : A 2, A 3 S 1 : A 3 φ 4 φ φ V : A 2, A 4 V 2 : A 2, A 3, A V 1 : A 3, A Junction tree propagation p. 1/2

45 Propagation in junction trees Propagation To calculate P(A i ) for any A i we send messages away from V. Example continued φ 1, φ 2, φ 3 V 4 : A 1, A 2, A 3 2 : ψ 4 : ψ 2 =... 4 : ψ 1 4 =... =... 1 : ψ 1 : ψ 1 = φ S 1 2 = φ S 2 3 : ψ 4 = φ A 2 4 S 4 : A 2 S 2 : A 2, A 3 S 1 : A 3 φ φ 4 φ V : A 2, A 4 V 2 : A 2, A 3, A V 1 : A 3, A ψ 2 = P A 1 ψ 1 φ 1 φ 2 φ 3 ψ 4 {ψ 4, P A 1 φ 1 φ 2 φ 3, ψ 1 } ψ 1 = P A 2 ψ 2 ψ 4 PA 1 φ 1 φ 2 φ 3 Junction tree propagation p. 1/2

46 Propagation in junction trees In general: Sending a message from C i to C j m li C l C j m ij C i m mi C m m ij := X Y Y φ m ki C j \C i φ Φ Ci C k nd(c i ) Junction tree propagation p. 19/2

47 Exact Inference Messages passing in general R Collect and distribute messages i) Collect messages to a preselected root R Junction tree propagation p. 20/2

48 Exact Inference Messages passing in general 1 1 R 1 Collect and distribute messages i) Collect messages to a preselected root R Junction tree propagation p. 20/2

49 Exact Inference Messages passing in general R 1 Collect and distribute messages i) Collect messages to a preselected root R Junction tree propagation p. 20/2

50 Exact Inference Messages passing in general R 3 1 Collect and distribute messages i) Collect messages to a preselected root R ii) Distribute messages away from the root R Junction tree propagation p. 20/2

51 Exact Inference Messages passing in general R Collect and distribute messages i) Collect messages to a preselected root R ii) Distribute messages away from the root R Junction tree propagation p. 20/2

52 Propagation in junction trees Theorem Let the junction tree T represent the Bayesian network BN over the universe U and with evidence e. Assume that T is full. 1. Let V be a clique with set of potentials Φ V, and let S 1,..., S k be V s neighboring separators and with V -directed messages Ψ 1,..., Ψ k. Then P(V, e) = Y Φ V Y Ψ1... YΨ k. 2. Let S be a separator with the sets Ψ S and Ψ S in the mailboxes. Then P(S, e) = Y Ψ S Y Ψ S. Junction tree propagation p. 21/2

53 Propagation in junction trees Theorem Let the junction tree T represent the Bayesian network BN over the universe U and with evidence e. Assume that T is full. 1. Let V be a clique with set of potentials Φ V, and let S 1,..., S k be V s neighboring separators and with V -directed messages Ψ 1,..., Ψ k. Then P(V, e) = Y Φ V Y Ψ1... YΨ k. 2. Let S be a separator with the sets Ψ S and Ψ S in the mailboxes. Then P(S, e) = Y Ψ S Y Ψ S. Evidence Evidence is inserted by adding corresponding 0/1-potentials to the appropriate cliques. Junction tree propagation p. 21/2

54 Triangulation of graphs So far we have assumed that the domain graph is triangulated. If this is not the case, then we embed it in a triangulated graph. Example Junction tree propagation p. 22/2

55 Triangulation of graphs So far we have assumed that the domain graph is triangulated. If this is not the case, then we embed it in a triangulated graph. Example A triangulated graph extending (b) Junction tree propagation p. 22/2

56 Triangulation of graphs Junction tree propagation p. 23/2

57 Triangulation of graphs A heuristic Repeatedly eliminate a simplicial node, and if this is not possible eliminate a node minimizing (fa(x) are noneliminated neighbors of X): fa(x) (fill-in-size ). Junction tree propagation p. 23/2

58 Triangulation of graphs 1 1,2,3 2,4, ,3, 2 3,, 1 2,,, A heuristic Repeatedly eliminate a simplicial node, and if this is not possible eliminate a node minimizing (fa(x) are noneliminated neighbors of X): fa(x) (fill-in-size ). Junction tree propagation p. 23/2

59 Triangulation of graphs 1 1,2,3 2,4, ,3, 2 3,, 1 2,,, A heuristic Repeatedly eliminate a simplicial node, and if this is not possible eliminate a node minimizing (fa(x) are noneliminated neighbors of X): fa(x) (fill-in-size ). Junction tree propagation p. 23/2

60 Triangulation of graphs 1,2,3 2,4, 2,3, 3,,,,, A heuristic Repeatedly eliminate a simplicial node, and if this is not possible eliminate a node minimizing (fa(x) are noneliminated neighbors of X): fa(x) (fill-in-size ). Junction tree propagation p. 23/2

61 Triangulation of graphs A heuristic Repeatedly eliminate a simplicial node, and if this is not possible eliminate a node minimizing (fa(x) are noneliminated neighbors of X): fa(x) (fill-in-size ). Junction tree propagation p. 23/2

62 Triangulation of graphs A heuristic Repeatedly eliminate a simplicial node, and if this is not possible eliminate a node minimizing (fa(x) are noneliminated neighbors of X): fa(x) (fill-in-size ). Junction tree propagation p. 23/2

63 Triangulation of graphs A heuristic Repeatedly eliminate a simplicial node, and if this is not possible eliminate a node minimizing (fa(x) are noneliminated neighbors of X): fa(x) (fill-in-size ). Junction tree propagation p. 23/2

64 Triangulation of graphs A heuristic Repeatedly eliminate a simplicial node, and if this is not possible eliminate a node minimizing (fa(x) are noneliminated neighbors of X): fa(x) (fill-in-size ). Junction tree propagation p. 23/2

65 Triangulation of graphs A heuristic Repeatedly eliminate a simplicial node, and if this is not possible eliminate a node minimizing (fa(x) are noneliminated neighbors of X): fa(x) (fill-in-size ). Junction tree propagation p. 23/2

66 Triangulation of graphs A heuristic Repeatedly eliminate a simplicial node, and if this is not possible eliminate a node minimizing (fa(x) are noneliminated neighbors of X): fa(x) (fill-in-size ). sp(fa(x)) (clique-size). P {Y,Z}:{Y,Z} nb(x) Z nb(y ) sp({y, Z}) (fill-in-weight) Junction tree propagation p. 23/2

67 Triangulation of graphs Usually there are many possible triangulations and finding the best one is NP-hard! Junction tree propagation p. 23/2

68 Summary Bayesian network Junction tree propagation p. 24/2

69 Summary Bayesian network Moral graph Junction tree propagation p. 24/2

70 Summary Bayesian network Moral graph Triangulated graph Junction tree propagation p. 24/2

71 Summary Bayesian network Moral graph Triangulated graph 1,2,3 2,4, 2,3, 3,,,,, Join graph Junction tree propagation p. 24/2

72 Summary Bayesian network Moral graph Triangulated graph 1,2,3 2,4, 1,2,3 2,4, 2,3, 3,, 2,3, 3,,,,, Join graph,,, Join tree Junction tree propagation p. 24/2

73 Summary Bayesian network Moral graph Triangulated graph 1,2,3 2,4, 1,2,3 2,4, 1,2,3 2,4, 2,3 2, 2,3, 3,, 2,3, 3,, 2,3, 3, 3,,,,,,,,,,,, Join graph Join tree Junction tree Junction tree propagation p. 24/2

74 Exact Inference Summary Given: Bayesian network BN for P(V) Junction tree propagation p. 2/2

75 Exact Inference Summary Given: Bayesian network BN for P(V) Moralize Domain graph G for BN (also called the moral graph) Junction tree propagation p. 2/2

76 Exact Inference Summary Given: Bayesian network BN for P(V) Moralize Domain graph G for BN (also called the moral graph) Triangulate Triangulated graph for G Junction tree propagation p. 2/2

77 Exact Inference Summary Given: Bayesian network BN for P(V) Moralize Domain graph G for BN (also called the moral graph) Triangulate Triangulated graph for G Find cliques Join graph Junction tree propagation p. 2/2

78 Exact Inference Summary Given: Bayesian network BN for P(V) Moralize Domain graph G for BN (also called the moral graph) Triangulate Triangulated graph for G Find cliques Join graph Find maximal spanning tree Join tree Junction tree propagation p. 2/2

79 Exact Inference Summary Given: Bayesian network BN for P(V) Moralize Domain graph G for BN (also called the moral graph) Triangulate Triangulated graph for G Find cliques Join graph Find maximal spanning tree Join tree Perform belief updating by message passing. Junction tree propagation p. 2/2