Graphical Models Another Approach to Generalize the Viterbi Algorithm

Transcription

1 Exact Marginalization Another Approach to Generalize the Viterbi Algorithm Oberseminar Bioinformatik am 20. Mai 2010 Institut für Mikrobiologie und Genetik Universität Göttingen 1.1

2 Undirected Graphical Model Undirected Graphical Model (Markov random field) describes joint probability distribution of a set of random variables x = (x v ) v V conditional independence properties of the random variables given by an undirected graph G = (V, E) p(x v x \v ) = p(x v x N(v) ) (for all v V ) Here: N(v) = {w V {v, w} E} = set of neighbours of v in the graph Example HMM / linear chain CRF x 1 x i 1 x i x i+1 x n P(x i x 1,..., x i 1, x i+1,..., x n ) = P(x i x i 1, x i+1 ) 1.2

3 Undirected Graphical Model Factorization (Hammersley and Clifford, 1971) All undirected graphical models with P(x) > 0 are of this form: P(x) = 1 Z ψ C (x C ) C C ψ C > 0: potential functions C: subsets of V (maximal cliques or subsets thereof) x C = (x v ) v C = variables associated with the subset C Z implicitly defined normalization constant 1.3

4 Undirected Graphical Model Factorization (Hammersley and Clifford, 1971) All undirected graphical models with P(x) > 0 are of this form: P(x) = 1 Z ψ C (x C ) C C ψ C > 0: potential functions C: subsets of V (maximal cliques or subsets thereof) x C = (x v ) v C = variables associated with the subset C Z implicitly defined normalization constant Example (HMM) Conditional on observation sequence y: n P(x y) = 1 P(y) P(y i x i )P(x i x i 1 ) i=1 Z = P(y), C = V E ψ i = P(y i x i ) (emission probs) ψ {i 1,i} = P(x i x i 1 ) (transition probs) 1.3

5 CRFs in Particular Conditional Random Field Distribution conditional on observation y is an undirected graphical model: P(x y) = 1 ψ C (x C, y) Z (y) C C 1.4

6 CRFs in Particular Conditional Random Field Distribution conditional on observation y is an undirected graphical model: P(x y) = 1 ψ C (x C, y) Z (y) C C exponential form In practice, potentials are of this exponential form k ψ C (x C, y) = exp w i f i (x C, y, C), where the feature functions f i are often binary features. Remark: HMMs are still of this form. i=1 1.4

7 (Conditional) Random Fields Quantities of Interest 1 most likely labeling arg max P(x y) (or approx.) corresponds to most likely gene structure most likely alignment denoised image best party plan x 1.5

8 (Conditional) Random Fields Quantities of Interest 1 most likely labeling arg max P(x y) (or approx.) corresponds to most likely gene structure most likely alignment denoised image best party plan 2 posterior probabilities / marginals: P(x W y) / P(x W ) for W a vertex or clique x 1.5

9 (Conditional) Random Fields Quantities of Interest 1 most likely labeling arg max P(x y) (or approx.) corresponds to most likely gene structure most likely alignment denoised image best party plan 2 posterior probabilities / marginals: P(x W y) / P(x W ) for W a vertex or clique x 3 normalization constant Z e.g. to actually compute P(x) for given x (classification) 1.5

10 Reminder: Had Developed Generalized Viterbi Algorithm Own algorithm Finds most likely element (labeling) when C = vertices and edges. ˆx arg max P(x), x H1 B1 B2 H2 H3 B3 Implementation Java program GeneralViterbi written by Moritz Maneke in Bachelor thesis. 1.6

11 Application: Comparative Gene Finding Danesh Morady Garavand (lab rotation) use GeneralViterbi as a new approach to comparative gene finding: Find genes simultaneously in two or more orthologous DNA sequences. Example (two aligned genomes) horizontal edges: neighboring bases in same species vertical edges: aligned bases in different species 1.7

12 Application: Comparative Gene Finding Comparative Model used simple label space with 7 labels {N,E0,E1,E2,I0,I1,I2} 1.8

13 Application: Comparative Gene Finding Comparative Model used simple label space with 7 labels {N,E0,E1,E2,I0,I1,I2} reused parameters of AUGUSTUS (linear chain CRF) for all node features and horizontal edge features (emission probabilities, transition probabilities) without vertical edges, two independent ab initio predictions 1.8

14 Application: Comparative Gene Finding Comparative Model used simple label space with 7 labels {N,E0,E1,E2,I0,I1,I2} reused parameters of AUGUSTUS (linear chain CRF) for all node features and horizontal edge features (emission probabilities, transition probabilities) without vertical edges, two independent ab initio predictions used very simple vertical edge features: f (x {u,v}, y, {u, v}) = { 1, if xu = x v 0, otherwise same labels of aligned bases are rewarded 1.8

15 Application: Comparative Gene Finding Results/Conclusions test on sequence pairs of length 5000 accuracy improvement of with vertical edges over without vertical edges : 8 percent points on exon level extension to production setting conceptionally easy, e.g. 12 Drosophila species, evidence integration 1.9

16 Application: Comparative Gene Finding Results/Conclusions test on sequence pairs of length 5000 accuracy improvement of with vertical edges over without vertical edges : 8 percent points on exon level extension to production setting conceptionally easy, e.g. 12 Drosophila species, evidence integration But: efficiency drops exponentially with width of graph 1.9

17 Application: Protein-Protein-Interaction interface residue backbone close in space Find labeling of vertices in {interface, not interface} based on vertex and edge features.

18 Limits of GeneralViterbi Problems GeneralViterbi too inefficient on very complex graphs (problem inherent limitation: NP-hard) also can use marginals like P(x v = l y) for training CRFs But: my approach not directly transferrable to marginalization Forward algorithm working, but not clear what corresponding Backwards variables would be. 1.11

19 Eliminate Limits Catch 2 Birds with 1 Stone: : 1 can do maximization and marginalization with very similar algorithm 2 can tackle complex graphs by running an approximation variant of the same algorithm 1.12

20 Eliminate Limits Catch 2 Birds with 1 Stone: : 1 can do maximization and marginalization with very similar algorithm 2 can tackle complex graphs by running an approximation variant of the same algorithm Maybe 3rd Bird: can use higher-order clique features in PPI. E.g. patch features. : most interfaces consist of [...] buried cores surrounded by partially accessible rims amino acid composition of the core differs considerably from the rim (Shoemaker and Panchenko, 2007) 1.12

21 Reminder: Definition () A tree decomposition of an undirected graph G = (V, E) is a tree (T, M), in which the vertices A T (bags) are subsets of vertices in G A = V A T u, v E A T with u, v A A, B T : C on the unique path from A to B A B C (running intersection property) Example 1.13 Copyright by Hein Röhrig

22 Tree Width of a Graph Definition (tree width) the width of a tree decomposition is: (size of largest bag) - 1 the tree width of a graph is the minimal width of all its tree decompositions 1.14

23 Compute Narrow For each k there is a linear time algorithm that tests whether a graph G = (V, E) has tree width at most k and, if so, computes a tree decomposition (T, M) of width at most k. Hans Bodlaender,

24 Compute Narrow For each k there is a linear time algorithm that tests whether a graph G = (V, E) has tree width at most k and, if so, computes a tree decomposition (T, M) of width at most k. Hans Bodlaender, 1992 Practial Solutions exakt k algorithm complicated, but linear time implementation is available: Diplomarbeit of Hein Röhrig, 1998, GNU licence,

25 Compute Narrow For each k there is a linear time algorithm that tests whether a graph G = (V, E) has tree width at most k and, if so, computes a tree decomposition (T, M) of width at most k. Hans Bodlaender, 1992 Practial Solutions exakt k algorithm complicated, but linear time implementation is available: Diplomarbeit of Hein Röhrig, 1998, GNU licence, junction tree: Can also add edges to E ( Triangulation), so that in the resulting graph G a tree of cliques exists that is a tree decomposition of G. 1.15

26 Compute Narrow For each k there is a linear time algorithm that tests whether a graph G = (V, E) has tree width at most k and, if so, computes a tree decomposition (T, M) of width at most k. Hans Bodlaender, 1992 Practial Solutions exakt k algorithm complicated, but linear time implementation is available: Diplomarbeit of Hein Röhrig, 1998, GNU licence, junction tree: Can also add edges to E ( Triangulation), so that in the resulting graph G a tree of cliques exists that is a tree decomposition of G. may chose to remove edges 1.15

27 Overview: Setting Have: undirected graphical model with potentials ψ C (x c ), tree decomposition of graph Want: marginal distributions P(x W ) for bags W of tree decomposition 1.16

28 Overview: Setting Have: undirected graphical model with potentials ψ C (x c ), tree decomposition of graph Want: marginal distributions P(x W ) for bags W of tree decomposition Remark: will give us posterior probs of vertices and edges needed for CRF training with IIS: P(x v y), P(x {v,w} y) 1.16

29 Overview: Setting Have: undirected graphical model with potentials ψ C (x c ), tree decomposition of graph Want: marginal distributions P(x W ) for bags W of tree decomposition Remark: will give us posterior probs of vertices and edges needed for CRF training with IIS: P(x v y), P(x {v,w} y) Idea: algorithm will maintain a distribution φ W for each bag ( potential ) 1.16

30 Overview: Setting Have: undirected graphical model with potentials ψ C (x c ), tree decomposition of graph Want: marginal distributions P(x W ) for bags W of tree decomposition Remark: will give us posterior probs of vertices and edges needed for CRF training with IIS: P(x v y), P(x {v,w} y) Idea: algorithm will maintain a distribution φ W for each bag ( potential ) will iteratively and locally update potentials, by passing messages between neighboring bags 1.16

31 Overview: Setting Have: undirected graphical model with potentials ψ C (x c ), tree decomposition of graph Want: marginal distributions P(x W ) for bags W of tree decomposition Remark: will give us posterior probs of vertices and edges needed for CRF training with IIS: P(x v y), P(x {v,w} y) Idea: algorithm will maintain a distribution φ W for each bag ( potential ) will iteratively and locally update potentials, by passing messages between neighboring bags after a fixed number of steps potentials are true marginals 1.16

32 Separators For each edge {X, Y } in the tree decomposition we add a separator node S between X and Y with all vertices in the intersection: S = X Y Example X S Y abcd cd cdef d dgh separator bag Why? 1.17

33 Separators For each edge {X, Y } in the tree decomposition we add a separator node S between X and Y with all vertices in the intersection: S = X Y Example X S Y abcd cd cdef d dgh separator bag Local Consistency When potentials are true marginals of P(x), then they agree on their common vertices. In particular, φ X and φ y must eventually agree on S. 1.17

34 Local Consistency Consistency on Separator We need X\S φ X = φ S = Y \S φ Y Example X S Y abcd cd cdef φ X (a, b, c, d) = φ S (c, d) = φ Y (c, d, e, f ) a,b e,f 1.18

35 Global Consistency Must have for any bags X, Y with intersection I, that φ X = φ Y. X\I Y \I 1.19

36 Global Consistency Must have for any bags X, Y with intersection I, that φ X = φ Y. X\I Y \I running intersection property & local consistency = global consistency 1.19

37 Messages Example φ X φ S φ Y X S Y 1.20

38 Messages Example φ X φ S φ Y X S Y Message from bag X to bag Y update φ S and φ Y according to φ S X\S φ X (1) φ Y φ Y φ S φ S (2) φ S φ S (3) 1.20

39 Two Passes: Leaves to Root to Leaves 1 initialize potentials: φ A = ψ A on bags, φ 1 on separators 1.21

40 Two Passes: Leaves to Root to Leaves 1 initialize potentials: φ A = ψ A on bags, φ 1 on separators 2 choose root bag arbitrarily 1.21

41 Two Passes: Leaves to Root to Leaves 1 initialize potentials: φ A = ψ A on bags, φ 1 on separators 2 choose root bag arbitrarily 3 pass messages from leaves to root (depth first search order) 1.21

42 Two Passes: Leaves to Root to Leaves 1 initialize potentials: φ A = ψ A on bags, φ 1 on separators 2 choose root bag arbitrarily 3 pass messages from leaves to root (depth first search order) 4 pass messages from root to leaves (reverse order) 1.21

43 Two Passes: Leaves to Root to Leaves 1 initialize potentials: φ A = ψ A on bags, φ 1 on separators 2 choose root bag arbitrarily 3 pass messages from leaves to root (depth first search order) 4 pass messages from root to leaves (reverse order) 5 potentials are exact marginals, Z can be obtained by summming any potential 1.21

44 Two Passes: Leaves to Root to Leaves 1 initialize potentials: φ A = ψ A on bags, φ 1 on separators 2 choose root bag arbitrarily 3 pass messages from leaves to root (depth first search order) 4 pass messages from root to leaves (reverse order) 5 potentials are exact marginals, Z can be obtained by summming any potential Example 1.21

45 Two Passes: Leaves to Root to Leaves 1 initialize potentials: φ A = ψ A on bags, φ 1 on separators 2 choose root bag arbitrarily 3 pass messages from leaves to root (depth first search order) 4 pass messages from root to leaves (reverse order) 5 potentials are exact marginals, Z can be obtained by summming any potential Example Forward pass: leaves to root 1.21

46 Two Passes: Leaves to Root to Leaves 1 initialize potentials: φ A = ψ A on bags, φ 1 on separators 2 choose root bag arbitrarily 3 pass messages from leaves to root (depth first search order) 4 pass messages from root to leaves (reverse order) 5 potentials are exact marginals, Z can be obtained by summming any potential Example Forward pass: leaves to root 1.21

47 Two Passes: Leaves to Root to Leaves 1 initialize potentials: φ A = ψ A on bags, φ 1 on separators 2 choose root bag arbitrarily 3 pass messages from leaves to root (depth first search order) 4 pass messages from root to leaves (reverse order) 5 potentials are exact marginals, Z can be obtained by summming any potential Example Forward pass: leaves to root 1.21

48 Two Passes: Leaves to Root to Leaves 1 initialize potentials: φ A = ψ A on bags, φ 1 on separators 2 choose root bag arbitrarily 3 pass messages from leaves to root (depth first search order) 4 pass messages from root to leaves (reverse order) 5 potentials are exact marginals, Z can be obtained by summming any potential Example Forward pass: leaves to root 1.21

49 Two Passes: Leaves to Root to Leaves 1 initialize potentials: φ A = ψ A on bags, φ 1 on separators 2 choose root bag arbitrarily 3 pass messages from leaves to root (depth first search order) 4 pass messages from root to leaves (reverse order) 5 potentials are exact marginals, Z can be obtained by summming any potential Example Forward pass: leaves to root 1.21

50 Two Passes: Leaves to Root to Leaves 1 initialize potentials: φ A = ψ A on bags, φ 1 on separators 2 choose root bag arbitrarily 3 pass messages from leaves to root (depth first search order) 4 pass messages from root to leaves (reverse order) 5 potentials are exact marginals, Z can be obtained by summming any potential Example Forward pass: leaves to root 1.21

51 Two Passes: Leaves to Root to Leaves 1 initialize potentials: φ A = ψ A on bags, φ 1 on separators 2 choose root bag arbitrarily 3 pass messages from leaves to root (depth first search order) 4 pass messages from root to leaves (reverse order) 5 potentials are exact marginals, Z can be obtained by summming any potential Example Forward pass: leaves to root 1.21

52 Two Passes: Leaves to Root to Leaves 1 initialize potentials: φ A = ψ A on bags, φ 1 on separators 2 choose root bag arbitrarily 3 pass messages from leaves to root (depth first search order) 4 pass messages from root to leaves (reverse order) 5 potentials are exact marginals, Z can be obtained by summming any potential Example Forward pass: leaves to root 1.21

53 Two Passes: Leaves to Root to Leaves 1 initialize potentials: φ A = ψ A on bags, φ 1 on separators 2 choose root bag arbitrarily 3 pass messages from leaves to root (depth first search order) 4 pass messages from root to leaves (reverse order) 5 potentials are exact marginals, Z can be obtained by summming any potential Example Backward pass: root to leaves 1.21

54 Two Passes: Leaves to Root to Leaves 1 initialize potentials: φ A = ψ A on bags, φ 1 on separators 2 choose root bag arbitrarily 3 pass messages from leaves to root (depth first search order) 4 pass messages from root to leaves (reverse order) 5 potentials are exact marginals, Z can be obtained by summming any potential Example Backward pass: root to leaves 1.21

55 Two Passes: Leaves to Root to Leaves 1 initialize potentials: φ A = ψ A on bags, φ 1 on separators 2 choose root bag arbitrarily 3 pass messages from leaves to root (depth first search order) 4 pass messages from root to leaves (reverse order) 5 potentials are exact marginals, Z can be obtained by summming any potential Example Backward pass: root to leaves 1.21

56 Two Passes: Leaves to Root to Leaves 1 initialize potentials: φ A = ψ A on bags, φ 1 on separators 2 choose root bag arbitrarily 3 pass messages from leaves to root (depth first search order) 4 pass messages from root to leaves (reverse order) 5 potentials are exact marginals, Z can be obtained by summming any potential Example Backward pass: root to leaves 1.21

57 Two Passes: Leaves to Root to Leaves 1 initialize potentials: φ A = ψ A on bags, φ 1 on separators 2 choose root bag arbitrarily 3 pass messages from leaves to root (depth first search order) 4 pass messages from root to leaves (reverse order) 5 potentials are exact marginals, Z can be obtained by summming any potential Example Backward pass: root to leaves 1.21

58 Two Passes: Leaves to Root to Leaves 1 initialize potentials: φ A = ψ A on bags, φ 1 on separators 2 choose root bag arbitrarily 3 pass messages from leaves to root (depth first search order) 4 pass messages from root to leaves (reverse order) 5 potentials are exact marginals, Z can be obtained by summming any potential Example Backward pass: root to leaves 1.21

59 Two Passes: Leaves to Root to Leaves 1 initialize potentials: φ A = ψ A on bags, φ 1 on separators 2 choose root bag arbitrarily 3 pass messages from leaves to root (depth first search order) 4 pass messages from root to leaves (reverse order) 5 potentials are exact marginals, Z can be obtained by summming any potential Example Backward pass: root to leaves 1.21

60 Two Passes: Leaves to Root to Leaves 1 initialize potentials: φ A = ψ A on bags, φ 1 on separators 2 choose root bag arbitrarily 3 pass messages from leaves to root (depth first search order) 4 pass messages from root to leaves (reverse order) 5 potentials are exact marginals, Z can be obtained by summming any potential Example 1.21 Have local consistency between X and Y after backward pass!

61 Party Planning Example binary classification of variables S, O, D, M, H: 1=invited, 0=not invited Stephan Oliver Dana BANG Marc Hamed P = 1 Z ψ S ψ O ψ Dψ Mψ Hψ SO ψ OD ψ MO ψ DH v : ψ v(0) = 1, ψ v(1) = 2 S O φ DH O D φ DH D H φ DH M O φ DH

62 Party Planning Example: Brute Force 1.23 distribution O D M S H sum 608 marginals P(S = 1) = 418/608 P(O = 1) = 380/608 P(D = 1) = 320/608 P(M = 1) = 152/608 P(H = 1) = 320/608

63 Party Planning Example: SO O OD D DH O MO 1.24

64 Party Planning Example: SO O OD D DH O MO P = 1 Z ψ SOψ OD ψ MO ψ DH define potentials ψ to achieve equivalence: (arbitrarily) assign original potentials ψ to bags S O φ SO O D φ OD D H φ DH M O φ MO

65 Party Planning Example S O φ SO O φ O O D φ OD D φ D D H φ DH SO O OD D DH O O φ O MO M O φ MO Initialize potentials: bags with original potentials ψ C separators with unity

66 Party Planning Example S O φ SO O φ O O D φ OD D φ D D H φ DH SO O OD D DH O O φ O MO M O φ MO choose root

67 Party Planning Example S O φ SO O φ O O D φ OD D φ D D H φ DH SO O OD D DH O O φ O MO M O φ MO pass messages from leaves to root

68 Party Planning Example S O φ SO O φ O O D φ OD D φ D D H φ DH SO O OD D DH O O φ O MO M O φ MO pass messages from leaves to root

69 Party Planning Example S O φ SO O φ O O D φ OD D φ D D H φ DH SO O OD D DH O O φ O MO M O φ MO pass messages from leaves to root

70 Party Planning Example S O φ SO O φ O O D φ OD D φ D D H φ DH Z 608 SO O OD D DH O O φ O MO M O φ MO

71 Party Planning Example S O φ SO O φ O O D φ OD Z 608 D φ D Z 608 D H φ DH Z 608 SO O OD D DH O O φ O MO M O φ MO pass messages from root to leaves

72 Party Planning Example S O φ SO O φ O O D φ OD Z 608 D φ D Z 608 D H φ DH Z 608 SO O OD D DH O O φ O Z 608 MO M O φ MO Z pass messages from root to leaves

73 Party Planning Example S O φ SO Z 608 O φ O Z 608 O D φ OD Z 608 D φ D Z 608 D H φ DH Z 608 SO O OD D DH O O φ O Z 608 MO M O φ MO Z pass messages from root to leaves

74 Party Planning Example S O φ SO Z 608 O φ O Z 608 O D φ OD Z 608 D φ D Z 608 D H φ DH Z 608 SO O OD D DH O O φ O Z 608 MO M O φ MO Z pass messages from root to leaves

75 Results We get... Z = 608 P(S = 1) = 418/608 P(O = 1) = 380/608 P(D = 1) = 320/608 P(M = 1) = 152/608 P(H = 1) = 320/608 just as in the brute force results. 1.26

76 Maximization Viterbi arg max P(x) x can be computed with a conceptually very similar algorithm in same running time. 1.27

77 Approximation rough procedure 1 use a graph of bags (that need be not a tree decomposition) 2 pass messages according to some schedule 3 stop passing messages at will 1.28

78 Literature Christopher M. Bishop: Pattern Recognition and Machine Learning, 2006 (book, chapter online available at cmbishop/prml) The Junction Tree Algorithm, script of Chris Williams, School of Informatics, University of Edinburgh, 2009 Graphical models, message-passing algorithms, and variational methods: Part I, Martin Wainwright,