Probabilistic Graphical Models (1): representation

Transcription

1 1 / 28 Probabilistic Graphical Models (1): representation Qinfeng (Javen) Shi The ustralian entre for Visual Technologies, The University of delaide, ustralia 15 pril 2011

2 ourse Outline Probabilistic Graphical Models: 1 Representation (Today) 2 Inference 3 Learning 4 Sampling-based approximate inference 5 Temporal models 6 2 / 28

3 History Gibbs (1902) used undirected graphs in particles Wright (1921,1934) used directed graph in genetics In economists and social sci (Wold 1954, lalock, Jr. 1971) In statistics (artlett 1935, Vorobev 1962, Goodman 1970, Haberman 1974) In I, expert system (ombal et al. 1972, Gorry and arnett 1968, Warner et al. 1961) Widely accepted in late 1980s. Prob Reasoning in Intelli Sys (Pearl 1988), Pathfinder expert system (Heckerman et al. 1992) Hot since RFs (Lafferty et al. 2001), SVM struct (Tsochantaridis etal 2004), M 3 Net (Taskar et al. 2004), DeepeliefNet (Hinton et al. 2006) 3 / 28

4 Good books hris ishop s book Pattern Recognition and Machine Learning (Graphical Models are in chapter 8, which is available from his webpage) 60 pages Koller and Friedman s Probabilistic Graphical Models > 1000 pages Stephen Lauritzen s Graphical Models Michael Jordan s unpublished book n Introduction to Probabilistic Graphical Models 4 / 28

5 Three main types of graphical models f1 f2 f3 (a) Directed graph (b) Undirected graph (c) Factor graph Nodes represent random variables. Edges represent dependencies between variables Factors explicitly show which variables are used in each factor i.e. f 1 (, )f 2 (, )f 3 (, ) 5 / 28

6 enefits of graphical models Relationships (and interactions) between variables are intuitive (such as conditional independences) compactly represent distributions of variables. have general inference algorithms (such as message-passing algorithms) to efficiently query P( = b, = c) or compute E P [f ] without enumerating all possible values of variables. 6 / 28

7 Independences and factorisation Independences give factorisation. Independence P(, ) = P()P() onditional Independence P(, ) = P( )P( ) 7 / 28

8 From graphs to factorisation 8 / 28 Directed cyclic Graph: P(x 1,..., x n ) = n i=1 P(x i Pa xi ) P(,, ) = P()P( )P(, )

9 From graphs to factorisation 9 / 28 Undirected Graph: P(x 1,..., x n ) = 1 Z c ψ c(x c ), Z = X c ψ c(x c ), where c is an index set of a clique (fully connected subgraph), X c is the set of variables indicated by c. P(,, ) = 1 Z ψ c 1 (, )ψ c2 (, )ψ c3 (, ), when X c1 = {, }, X c2 = {, }, X c3 = {, } or P(,, ) = 1 Z ψ c(,, ), when X c = {,, }

10 From graphs to factorisation 10 / 28 Factor Graph: P(x 1,..., x n ) = 1 Z i f i(x i ), Z = X i f (X i) f1 f2 f3 P(,, ) = 1 Z f 1(, )f 2 (, )f 3 (, )

11 From graphs to independences 11 / 28 ase 1: is said to be tail-to-tail. Head Tail Question:?

12 From graphs to independences ase 1: Question:? nswer: No. P(, ) = = P(,, ) P( )P( )P() P()P() in general 12 / 28

13 From graphs to independences 13 / 28 ase 1: Question:?

14 From graphs to independences ase 1: Question:? nswer: Yes. P(, ) = P(,, ) P() = P( )P( )P() P() = P( )P( ) 14 / 28

15 From graphs to independences 15 / 28 ase 2: is said to be head-to-tail. Question:,?

16 From graphs to independences 16 / 28 ase 3: is said to be head-to-head. Question:,?

17 From graphs to independences 17 / 28 ase 3: Question:,? P(,, ) = P()P()P(, ), P(, ) = P(,, ) = P()P()P(, ) = P()P()

18 D-separation - def Graph G(V, E) and nonintersecting sets X, Y, O V. How to check X Y O just by reading the graph G? onsider all paths from any node X to any node Y. path is said to be blocked by O, if it includes a node such that either exists a node O is either head-to-tail or tail-to-tail. does not exist a head-to-head node O, nor any of its descendants O. If all paths from X to Y are blocked by O, then X is said to be d-separated (directed separated) from Y by O. 18 / 28

19 D-separation - Example 19 / 28 D D E E F F Questions: Is F? heck if is d-separated from F by. Is F D? heck if is d-separated from F by D.

20 Inference - variable elimination 20 / 28 What is P(), or argmax,, P(,, )? P() =, P()P()P(, ) = P() P()P(, ) = P()m 1 (, ) ( eliminated) = m 2 () ( eliminated)

21 Inference - variable elimination 21 / 28 X1 X2 X3 P(x 1, x 2, x 3 ) = 1 Z ψ(x 1, x 2 )ψ(x 1, x 3 )ψ(x 1 )ψ(x 2 )ψ(x 3 ) P(x 1 ) = 1 ψ(x 1, x 2 )ψ(x 1, x 3 )ψ(x 1 )ψ(x 2 )ψ(x 3 ) Z x 2,x 3 = 1 Z ψ(x 1) ) ( ) (ψ(x 1, x 2 )ψ(x 2 ) ψ(x 1, x 3 )ψ(x 3 ) x 2 = 1 Z ψ(x 1)m 2 1 (x 1 )m 3 1 (x 1 ) x 3

22 Inference - variable elimination 22 / 28 X1 X2 X3 P(x 2 ) = 1 Z ψ(x 2) (ψ(x 1, x 2 )ψ(x 1 ) ) [ψ(x 1, x 3 )ψ(x 3 )] x 1 x 3 = 1 Z ψ(x 2) x 1 ψ(x 1, x 2 )ψ(x 1 )m 3 1 (x 1 ) = 1 Z ψ(x 2)m 1 2 (x 2 )

23 Inference - Message Passing 23 / 28 In general, P(x i ) = 1 Z ψ(x i) j Ne(i) m j i (x i ) m j i (x i ) = x j (ψ(x j )ψ(x i, x j ) k Ne(j)\{i} ) m k j (x j )

24 Inference - sum-product X1 X1 m2->1(x1) m1->2(x2) m3->2(x2) X2 m4->2(x2) m2->3(x3) X2 m2->4(x4) X3 X4 X3 X4 P(x i ) = 1 Z ψ(x i) j Ne(i) m j i (x i ) m j i (x i ) = x j (ψ(x j )ψ(x i, x j ) k Ne(j)\{i} ) m k j (x j ) called sum-product algorithm or belief propagation. 24 / 28

25 Inference - max-product To compute (x1,, x 4 ) = argmax x1,,x 4 use max-product algorithm. P(x 1,, x 4 ), X1 X1 m2->1(x1) m1->2(x2) m3->2(x2) X2 m4->2(x2) m2->3(x3) X2 m2->4(x4) X3 X4 X3 X4 x i ( = argmax ψ(x i ) ) m j i (x i ) x i j Ne(i) ( m j i (x i ) = max ψ(x j )ψ(x i, x j ) x j k Ne(j)\{i} ) m k j (x j ) 25 / 28

26 Inference - Message Passing in Log Space 26 / 28 To avoid over/underflow, log P(x i ) = log(ψ(x i )) + µ j i (x i ) log(z ) µ j i (x i ) := log m j i (x i ) j Ne(i)

27 real application Denoising 1 pplications in Vision and PR Image denoising Ψ x i, x ) ( j Real pplications Denoising Φ x i, y ) ( i Original Noisy orrected X = argmax X P(X Y ) 1 This example is from Tiberio aetano s short course: Machine Learning using Graphical Models 27 / 28

28 More details of P and other inference methods will be covered at the next talk. 28 / 28