Probabilistic Graphical Models

Transcription

1 Probabilistic Graphical Models Lecture 9 Undirected Models CS/CNS/EE 155 Andreas Krause

2 Announcements Homework 2 due next Wednesday (Nov 4) in class Start early!!! Project milestones due Monday (Nov 9) 4 pages of writeup, NIPS format Best project award!! 2

3 Answering multiple queries X 1 X 2 X 3 X 4 X 5 3

4 Reusing computation X 1 X 2 X 3 X 4 X 5 4

5 Next Will learn about algorithm for efficiently computing all marginalsp(x i E=e) given fixed evidence E=e Need appropriate data structure for storing the computation Junction trees 5

6 Junction trees C CD A junction tree for a collection of factors: D I DIG A tree, where each node is a cluster of variables H G L J S GIS GJSL HGJ JSL Every factor contained in some cluster C i Running intersection property: If X C i and X C j, and C m is on the path between C i and C j, then X C m 6

7 VE constructs a junction tree X 1 X 2 X 3 X 4 X 5 One clique C i for each factor f i created in VE C i connected to C j if f i used to generate f j Every factor used only once Tree Theorem: resulting tree satisfies RIP 7

8 Constructing JT from chordalgraph C D I G S L H J 8

9 Junction trees and independence CD DIG GIS GJSL HGJ JSL Theorem: Suppose T is a junction tree for graph G and factors F Consider edge C i C j with separator S i,j Variables Xand Yon opposite sites of separator Then X Y S i,j Furthermore, I(T) I(G) 9

10 VE as message passing CD DIG GIS GJSL HGJ C D I VE for computing X i Pick root (any clique containing X i ) G S Don t eliminate, only send messages recursively from leaves to root L Multiply incoming messages with clique potential H J Marginalize variables not in separator Root ready when received all messages 10

11 Correctness of message passing CD DIG GIS GJSL HGJ Theorem: When root ready (received all messages), all variables in root have correct potentials Follows from correctness of VE So far, no gain in efficiency 11

12 Does the choice of root affect messages? 1: CD 2: DIG 3: GIS 4: GJSL 5: HGJ 1: CD 2: DIG 3: GIS 4: GJSL 5: HGJ 12

13 Shenoy-Shafer algorithm C 1 C 2 C 4 Clique i ready if received messages from all neighbors Leaves always ready While there exists a message δ i j ready to transmit send message C 3 C 5 C 6 Complexity? Theorem: At convergence, every clique has correct beliefs 13

14 Inference using VE Want to incorporate evidence E=e Multiply all cliques containing evidence variables with indicator potential 1 e Perform variable elimination 14

15 Summary so far Junction trees represent distribution Constructed using elimination order Make complexity of inference explicitly visible Can implement variable elimination on junction trees to compute correct beliefs on all nodes Now: Belief propagation an important alternative to VE on junction trees. Will later generalize to approximate inference! Key difference: Messages obtained by division rather than multiplication 15

16 Message passing by factor division Variable elimination: Message Belief 1: CD 2: DIG Factor division: Belief Message 3: GIS 16

17 Factor division 17

18 Clique and separator potentials 1: AB 2: BC 18

19 Lauritzen-Spiegelhalter algorithm (a.k.a. Belief Propagation) C 1 Initialize separator potentials µ ij One per edge, initialized to 1 Messages C 2 C 4 C 3 C 5 C 6 19

20 Correctness of Belief propagation Complexity linear in #cliques Theorem: At convergence, every clique has correct beliefs (when using correct message order, i.e., leaves to root and back) Corollary: Junction tree is calibrated (cliques agree on separator) 20

21 Comparison of VE and BP messages Variable elimination C 1 C 3 C 4 C 2 Belief propagation C 1 C 3 C 4 C 2 21

22 Junction tree potential Understanding BP Junction tree invariant Theorem: BP maintains Junction tree invariant BP reparametrizes clique and separator potentials 22

23 Advantages and disadvantages of junction tree inference Advantages Can answer multiple queries (for same evidence) efficiently Can perform incremental updates Disadvantages No factors are deleted Can t prune away unnecessary variables Slower for a single query 23

24 Bayesian Networks Summary so far Representation Learning (MLE/ Bayesian) with fully observed data Exact Inference Next Undirected models Approximate inference Hidden variables 24

25 Representing the world using BNs s 4 s 1 s 2 s 3 represent s 5 s7 s s 8 6 s 11 s 9 s 10 True distribution P with cond. ind. I(P ) Want to make sure that I(P) I(P ) Ideally: I(P) = I(P ) Want BN that exactly captures independencies in P! s 12 Bayes net (G,P) with I(P) 25

26 Perfect maps Minimal I-maps are easy to find, but can contain many unnecessary dependencies. A BN structure G is called P-map(perfect map) for distribution P if I(G) = I(P) Does every distribution P have a P-map? 26

27 Existence of perfect maps Will have undirected model as P-map 27

28 Undirected Parameterization X 1 X 2 X 3 X 4 28

29 Markov Networks (a.k.a., Markov Random Field, Gibbs Distribution, ) A Markov Network consists of An undirected graph, where each node represents a RV A collection of factors defined over cliques in the graph Joint probability X 1 X 2 X 4 X 3 X 5 X 6 X 7 X 8 X 9 A distribution factorizes over undirected graph G if 29

30 Example MN: Image denoising Markov Network Y 1 Y 2 Y 3 X 1 X 2 X 3 Y 4 Y 5 Y 6 X 4 X 5 X 6 Y 7 Y 8 Y 9 X 7 X 8 X 9 X i : noisy pixels Y i : true pixels 30 30

31 Computing Joint Probabilities Computing joint probabilities in BNs Computing joint probabilities in Markov Nets 31

32 Independences in Markov Nets? In Bayes Nets (G,P) Local Markov Assumption: X NonDesc(X) Pa X G is I-map for distribution P if Local Markov Assumption holds Factorization Thm: P factorizes over G G is an I-map Global independences: d-separation Completeness and soundness of d-separation How about Markov Nets? What s the analog of the Local Markov Assumption? Is there a factorization theorem for Markov Nets? What replaces d-separation? 32

33 Local Markov Assumption for MN X 3 X 1 X 2 X 4 X 5 X 7 X 8 X 6 The Markov Blanket MB(X) of a node X is the set of neighbors of X Local Markov Assumption: X EverythingElse MB(X) I loc (G) = set of all local independences G is called an I-map of distribution P if I loc (G) I(P) X 9 33

34 Factorization Theorem for Markov Nets s 4 s 1 s 2 s 3 s 6 s 5 s7 s 8 s 11 s 9 s 10 True distribution P can be represented exactly as a Markov net (G,P) s 12 I loc (G) I(P) G is an I-mapof P (independence map) 34

35 Factorization Theorem for Markov Nets s 4 s 1 s 2 s 3 s 6 s 5 s7 s 8 s 11 s 9 s 10 s 12 I loc (G) I(P) True distribution P can be represented exactly as G is an I-mapof P (independence map) i.e., P can be represented as a Markov net (G,P) 35

36 Counterexample G an I-map for P does not imply that P factorizes Binary variables X 1,,X 4. Only positive states (0,0,0,0), (1,0,0,0), (1,1,0,0), (1,1,1,0) (0,0,0,1), (0,0,1,1), (0,1,1,1), (1,1,1,1) 36

37 Factorization Theorem for Markov Nets Hammersley-Clifford Theorem s 4 s 1 s 2 s 3 s 6 s 5 s7 s 8 s 11 s 9 s 10 s 12 I loc (G) I(P) True distribution P can be represented exactly as G is an I-mapof P (independence map) and P>0 i.e., P can be represented as a Markov net (G,P) 37

38 Global independencies X 1 A trail X X 1 X m Y is called active for evidence E, if none of X 1,,X m E X 3 X 2 X 4 X 5 X 6 Variables X and Y are called separatedby Eif there is no active trail for Econnecting X, Y Write sep(x,y E) X 7 X 8 I(G) = {X Y E: sep(x,y E)} X 9 38

39 Soundness of separation Know: For positive distributions P>0 I loc (G) I(P) P factorizes over G Theorem: Soundness of separation For positive distributions P>0 I loc (G) I(P) I(G) I(P) Hence, separation captures only true independences How about I(G) = I(P)? 39

40 Completeness of separation Theorem: Completeness of separation I(G) = I(P) for almost all distributions P that factorize over G almost all : Except for of potential parameterizations of measure 0 (assuming no finite set have positive measure) 40

41 Independences in Markov Nets? In Bayes Nets (G,P) Local Markov Assumption: X NonDesc(X) Pa X G is I-map for distribution P if Local Markov Assumption holds Factorization Thm: P factorizes over G G is an I-map Global independences: d-separation Completeness and soundness of d-separation How about Markov Nets? Local Markov Assumption: X EverythingElse MB(X) Factorization Thm: For positive P, P factorizes G is an I-map Global independences: separation For positive P: separation is complete and sound How about minimal I-maps and P-maps?? 41

42 Minimal I-maps For BNs: Minimal I-map not unique E B E B J M A A E B J M J M A For MNs: For positive P, minimal I-map is unique!! 42

43 Do P-maps always exist? For BNs: no P-maps How about Markov Nets? 43

44 Exact inference in MNs Variable elimination and junction tree inference work exactly the same way! Need to construct junction trees by obtaining chordalgraph through triangulation 44

45 Pairwise MNs A pairwise MN is a MN where all factors are defined over single variables or pairs of variables Can reduce any MN to pairwise MN! X 1 X 2 X 4 X 3 X 5 45

46 Tasks Read Koller& Friedman Chapters 10 and