Chapter 8 Cluster Graph & Belief Propagation. Probabilistic Graphical Models 2016 Fall

Transcription

1 Chapter 8 Cluster Graph & elief ropagation robabilistic Graphical Models 2016 Fall

2 Outlines Variable Elimination 消元法 imple case: linear chain ayesian networks VE in complex graphs Inferences in HMMs and linear-chain CRFs Exact Inference: Clique ree Cluster graph and clique tree Message passing: sum product Message passing: belief update Constructing clique tree

3 Outlines From clique tree to loopy cluster graph ethe cluster graph or cluster graph elief propagation as variational inference Extensions of belief propagation Generalized belief propagation Convex belief propagation Expectation propagation...

4 imple Case: VE in Chains C E e a b c d e d c b a

5 imple Case: VE in Chains y the chain rule for probabilities: C E d c b a d c b a d e c d b c a b a e d c b a e

6 imple Case: VE in Chains Rearranging terms... C E d c b a d e c d b c a b a e d c b a a b a d e c d b c

7 imple Case: VE in Chains erform the innermost summation C E d c b d c b a b p d e c d b c a b a d e c d b c e

8 imple Case: VE in Chains Rearrange and then sum again C E d c d c b d c b c p d e c d b p b c d e c d b p d e c d b c e

9 General ayesian Networks Visit moking uberculosis ung Cancer bnormality in Chest ronchitis -Ray yspnea Example: a simplified lung disease diagnostic network

10 V V V We want to compute Need to eliminate: V. V V V Initial factors

11 V Eliminate: V Note: τ 1 = Compute: V V V V 1. V V V. 1 We want to compute Need to eliminate: V Current factors

12 V Eliminate: Compute: 2 V We want to compute Need to eliminate: Current factors

13 V Eliminate: Compute: Note: τ 3 = 1 for all values of!! We want to compute Need to eliminate: Current factors

14 V Eliminate: Compute: We want to compute Need to eliminate: Current factors

15 V Eliminate: Compute: We want to compute Need to eliminate: Current factors

16 V Eliminate: Compute: We want to compute Need to eliminate: Current factors

17 We want to compute Need to eliminate: V Current factors 6 Eliminate: Compute: 7 6 Note: τ 7 is

18 ealing with Evidence V How do we deal with evidence? uppose get evidence V = v = s = d We want to compute V = v = s = d

19 ealing with Evidence Compute V = v = s = d Initial factors after setting evidence: V d s s v s v V ~ ~ ~ ~ ~ ~ V

20 Induced Graph in VE V Want to compute tep 1: Moralizing V

21 Want to compute Moralizing Eliminating V V V V V V V 1 Induced Graph in VE

22 Want to compute Moralizing Eliminating V Eliminating V V 2 V Induced Graph in VE

23 Induced Graph in VE Want to compute V Moralizing Eliminating V Eliminating Eliminating V 3

24 V Induced Graph in VE Want to compute Moralizing Eliminating V Eliminating Eliminating Eliminating V 5 2 4

25 Induced Graph in VE V Want to compute Moralizing Eliminating V Eliminating Eliminating Eliminating Eliminating V

26 V Induced Graph in VE Want to compute Moralizing Eliminating V Eliminating Eliminating Eliminating Eliminating Eliminating V

27 Induced Graph in VE 1 Moralized for N 2 Chordal V

28 VE: Inferences in HMMs and CRFs lease recall the graphic representations of HMMs MEMMs and linear-chain CRFs Given the backbone of Y is the same Y 0 Y 1 Y 2 Y 3 Y 0 Y 1 Y 2 Y

29 Compute θ VE: Forward lgorithm Initialization: i x... 1 x y i t t t Y 1 Y 2 Y 3 Induction: i e i i x N t 1 i t j t j i ei xt ermination: j1 1 N i1 i

30 VE isadvantages We need to traverse the whole graph for each run of reference Many intermediate results during previous runs of variable elimination can be re-used For example if we have run the reference of to infer results for eliminating V can be re-used We can directly re-start from V

31 VE isadvantages For the induced undirected graph of N moralized & chordal the basic structures are the cliques maximal C If we can pre-calculate the marginal distributions defined on maximal cliques the inferences may save many re-calculations G I Can we design an algorithm H J to achieve this goal?

32 Clique ree: Concrete Example Clique ree For a chordal graph tree-like structure by cliques maximal C I wo important features: G ree and family preserving Running intersection property H J C GI GI G GJ GI GJ HGJ

33 Clique ree: Concrete Example ree and family preserving Factors φ i are defined on cliques C Edges are defined on the sepset ij of two directly connected cliques G I Running intersection property ny variable only exists in a unique subpath along the tree H J C GI GI G GJ GI GJ HGJ

34 Clique ree: Concrete Example ssign local Cs to factors C I he clique tree is an equivalent representation of as the original factorization representation H G J C GI GI G GJ GI GJ HGJ C G I I G H GJ C I J

35 Exact Inference: Clique rees Exploits factorization of the distribution for efficient inference similar to variable elimination dvantage: avoid unnecessary or repeated computations if repeated queries are needed istribution un-normalized can be represented by clique tree with associated factors Φ = φ i i φ i Φ For ayesian networks factors are local Cs For Markov networks factors are clique potentials

36 Message assing: um roduct on Clique ree Goal: Compute J et initial factors at each cluster as products C 1 : Eliminate C sending 12 to C 2 C 2 : Eliminate sending 23 GI to C 3 C 3 : Eliminate I sending 35 G to C 5 C 4 : Eliminate H sending 45 GJ to C 5 C 5 : Obtain J by summing out G C H G I J C GI GI G GJ GI GJ HGJ C G I I G H GJ C I J

37 Message assing: um roduct on Clique ree Goal: Compute J et initial factors at each cluster as products C 1 : Eliminate C sending 12 to C 2 C 2 : Eliminate sending 23 GI to C 3 C 3 : Eliminate I sending 35 G to C 5 C 5 : Eliminate sending 54 GJ to C 4 C 4 : Obtain J by summing out HG C H G I J C GI GI G GJ GI GJ HGJ C G I I G H GJ C I J

38 C I G H J C GI GI G GJ GI GJ HGJ C G I I G H GJ C I J

39 Clique ree Message assing et be a clique tree and C 1...C k its cliques Multiply factors of each clique resulting in initial potentials as each factor is assigned to some clique then we have [ 0 j C j ] : j efine C r as the root cluster j 1 tart from tree leaves and move inward k [ 0 j C j ] C H G I J C GI GI G GJ GI GJ HGJ

40 Clique ree Message assing: Example Root C 6 egal ordering I: egal ordering II: Illegal ordering: C 1 C 4 C 6 C 5 C 3 C 2

41 Clique ree Calibration Calibration 校准 : the sending messages of two adjacent cliques should be equal For calculating the probability of any variable we need a more efficient algorithm to do the calculations rather than repeating above sum operations for each clique Obviously there are some information during message passing which can be re-used to calculate the probability of other variables C GI GI G GJ GI GJ HGJ

42 Clique ree Calibration We say that cluster C i is ready to transmit to neighbor C j when C i has messages from all its neighbors except C j. When C i is ready it can compute the message δ i j ij by multiplying its initial potential β i 0 final potential β i with all the coming messages δ k *Nbi j+ i and then eliminate the variables not in the sepset C i ij

43 Clique ree Calibration: Example Root: C 5 first downward pass C fter the upward pass we already get the marginal factor βgj G I H J C GI GI G GJ GI GJ HGJ C G I I G H GJ C I J

44 Clique ree Calibration: Example Root: C 5 second downward pass C I G H J C GI GI G GJ GI GJ HGJ C G I I G H GJ C I J

45 Clique ree Calibration: um-roduct GI G GJ C GI GI GJ HGJ GI G GJ C GI GI GJ HGJ

46 Clique ree Calibration: um-roduct GI G GJ C GI GI GJ HGJ GI G GJ C GI GI GJ HGJ fter calibration the final factors associated with cliques are updated to the marginal distributions or factors over the cliques

47 Clique ree Calibration If appears in multiple cliques they must agree clique tree with potentials i [C i ] is said to be calibrated if for all neighboring cliques C i and C j : Ci ij β i C i = Cj ij β j C j dvantage: compute posteriors for all the cliques using only twice passes

48 Calibrated Clique ree as istribution calibrated clique tree is more than simply a data structure that stores the results of probabilistic inference for all of the clique in the tree. It can also be viewed as an alternative representation of the measure φ un-normalized distribution. We can easily prove: he product of the marginal distributions or factors of all the cliques divided by the marginal factors of all the sepsets i C i j C C i C i i j i j

49 C C ayesian network Clique tree For calibrated tree Joint distribution can thus be written as C C C C C C C ] [ ] [ ] [ ] [ ] [ ] [ ] [ ] [ C C C Calibrated trees can be alternative representations of the distributions. hey are equal! Calibrated Clique ree as istribution

50 o Random-Order Message assing In the above sum-product message passing a factor i can transmit a message to j only if it has received all the other messages factor is ready Can the factor i transmit a message to its neighbors when it is not ready?

51 Message assing: elief Update Recall the clique tree calibration algorithm Upon calibration the final potential at i is: message from i to j sums out the non-sepset variables from the product of initial potential and all other messages Can also be viewed as multiplying all messages and dividing by the message from j to i i N i k i i k 0 } { 0 j N k i k C j i i j i i i i j C i i j N k i k C j i j i i i j i i i 0

52 Message assing: elief Update ayesian network Clique tree Root: C 2 C 1 to C 2 Message: C 2 to C 1 Message: lternatively compute nd then: hus the two approaches are equivalent ] [ ] [ ] [ ] [ ] [ ] [ ] [

53 Message assing: elief Update ased on the observation above belief update ifferent message passing scheme Each clique C i maintains its fully updated beliefs i product of initial messages and messages from neighbors tore at each sepset ij the previous message ij passed regardless of the direction When passing a message divide by previous ij Claim: message passing is correct regardless of the clique that sent the last message his is called belief update

54 lgorithm for elief Update he U is arbitrary. You can randomly choose any edge in the clique graph for the update. t convergence: β i C i ij = β j C j ij

55 Clique ree Invariant for U U maintains distribution invariant property Upon calibration we have C i[ C i ] i U i j i j Initially this invariant holds obviously t each update step invariant is also maintained Message only changes j and ij C C i j We need to prove β j U U μ ij = β j μ ij his is exactly the message passing step β U j = β j μ U ij μ ij elief update re-parameterizes at each step

56 Inference on Calibrated Clique rees ingle variable inference he posterior of a target variable can be directly computed by eliminating the redundant variables from a clique that contains Inference outside a clique Inference with increment or evidence fter calibration the final factors associated with cliques are updated to the marginal distributions or factors over the cliques

57 nswering Queries Outside a Clique Y ' iv ' ij ' i j Y Y ' cope ' i Find a minimal sub-path on the clique tree which contains all the query variables!

58 nswering Queries with Increments Introducing evidence Z=z Compute posterior of where is in a clique with Z ince clique tree is calibrated multiply the clique that contains and Z with indicator function IZ=z and sum out irrelevant variables Compute posterior of if not sharing a clique with Z Introduce indicator function IZ=z into some clique containing Z and propagate messages along path to clique containing um out irrelevant factors from clique containing

59 Comments on Calibrated Clique rees What is a calibrated clique tree? From the initial factors ψ C i or ψ i generated from Ns/MNs a clique tree is calibrated if we get the joint distributions β C i or β i associated with all nodes cliques and the joint distributions μ ij of all sepsets in the tree. We can use either sum product or belief update to do the calibration.

60 Comments on Calibrated Clique rees What is a calibrated clique tree? Why does a clique tree need to be calibrated? ll Cs/factors in Ns/MNs are equivalently transformed as calibrated factors and messages in clique tree. he joint distribution is invariant for the calibrated beliefs and all the steps of U. he calibrated factors β i and messages μ ij are directed and completely associated the clique tree.

61 Comments on Calibrated Clique rees What is a calibrated clique tree? Why does a clique tree need to be calibrated? What is the advantage of clique tree? In most cases the structure of clique tree is simpler than the original Ns/MNs. Inference will be more efficient. elief propagation can be easily extended to approximate inference

62 Constructing Clique rees Goal: construct a tree that is family preserving and obeys the running intersection property riangulate the graph to construct a chordal graph H N-hard to find triangulation where the largest clique in the resulting chordal graph has minimum size Find cliques in H and make each a node in the graph Finding maximal cliques is N-hard Can start with families and grow greedily Construct a tree over the clique nodes Use maximum spanning tree on an undirected graph whose nodes are maximal cliques and edge weight is C i C j Can show that resulting graph obeys running intersection

63 C C C I Moralized Graph I One possible triangulation I G G G H J H J H J C Cluster graph with edge weights GI GI G J GH Find the maximum spanning tree C GI GI G J GH 1 1

64 imitations of Clique ree Hard to construct induced graph & clique tree in large graph Inefficiency for Markov networks with loops airwise Markov Network

65 From Clique ree to oopy Cluster Graph Can we directly define clusters on the cliques of the original rather than induced graph? he message passing strategy cannot stop due to the problem of message circulating Markov Network Clique ree oopy Cluster Graph

66 in oopy Cluster Graph Markov Network oopy Cluster Graph

67 in oopy Cluster Graph pproximate marginal by multiplying initial beliefs and all the incoming messages new message by multiplying initial beliefs and all the incoming messages except i

68 Comments for in oopy Graph Initialize all the messages as 1 so any cluster can send its messages at the beginning on t require messages are equal in both directions Maintain the factor beliefs by multiplying initial beliefs/potentials with all the incoming messages end messages by multiplying initial beliefs/potentials with all the incoming messages except the target cluster eliminating the beliefs of the variables not in the sepset

69 Running Intersection roperty For any variable if a variable exists in cluster C i and C j there exists only a single path between the two clusters all the clusters on the path contain the variable If a cluster graph follows running intersection property the calibrated beliefs are approximate marginal potentials of the clusters eliefs do not change over time Or β i Ci ij = β j Cj ij

70 ethe Cluster Graph or Factor Graph wo types of nodes Factor nodes defined on cliques Variable nodes defined on variables Factor graph ensures running interaction property wo types of messages From factors to variables simply eliminate other variables From variables to factors C C C

71 ethe Cluster Graph or Factor Graph From variables to factors xf cx cnb x \ f From factors to variables f x eliefs after the propagation f x cf y cnb f y x x c x cnb x \ x C C Factor graph can easily calculate the approximate marginal of single variable C

72 he roblem of Convergence he first concern of belief propagation is the convergence If converged another concern is that the calibrated beliefs are not equal to the correct marginal potentials o we have some heuristic solutions?

73 he roblem of Convergence everal improvements Message scheduling: residual belief propagation order the messages according to their changes Minimum spanning tree: each time select a different spanning tree of the cluster graph and do calibration... Can we find a theoretical explanation of belief propagation on cluster graph?

74 n application to a 11*11 Ising model

75 as Variational Inference he major advance of the theoretical analysis of belief propagation is that Yedidia et al shows these approaches are maximizing an approximate energy functional his result connected the algorithmic developments in the field with literature on freeenergy approximations developed in statistical mechanics ethe 1935; Kikuchi 1951 such as mean field inference

76 Exact Inference as Optimization Energy functional K Q Φ = ln Z H Q + E Q φ φ Φ he second term is energy functional F Φ Q For a clique tree with clique C i we have a set of beliefs Q β i μ ij - not calibrated. Its energy functional: Φ = i ψ i ψ i is the initial factor for each clique F Φ Q = i H βi C i i H μij ij + i E βi ln ψ i

77 Exact Inference as Optimization If Q is calibrated as Q* we can conclude that F Φ Q = max Q F Φ Q = ln Z ecause the distribution is invariant for any calibrated beliefs the relative entropy is minimized as 0 ransform /U algorithm as optimization

78 Extensions of elief ropagation Generalized belief propagation Convex belief propagation Expectation propagation...

79 Region Graph

80 References for elief ropagation extbook #2: Chapter 22. More variational inference Wainwright MJ Michael IJ. Graphical models exponential families and variational inference. Foundations and rends in Machine earning 11-2:1-305.

81 he End of Chapter 8 Cluster graph is another data structure for efficient inferences of GMs