Recap: Inference in Probabilistic Graphical Models. R. Möller Institute of Information Systems University of Luebeck

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1 Recap: Inference in Probabilistic Graphical Models R. Möller Institute of Information Systems University of Luebeck

2 A Simple Example P(A,B,C) = P(A)P(B,C A) = P(A) P(B A) P(C B,A) = P(A) P(B A) P(C B) C is conditionally independent of A given B Graphical Representation???

3 Directed Graphical Model U = (V 1,, V n ) Bayesian Network P(U) = Õ P(V i Pa(V i )) P(A,B,C) = P(A) P(B A) P(C B) A B C

4 Digression: Polytrees A network is singly connected (a polytree) if it contains no undirected loops. D C Theorem: Inference in a singly connected network can be done in linear time*. Main idea: in variable elimination, need only maintain distributions over single nodes. 4 * in network size including table sizes. Jack Breese (Microsoft) & Daphne Koller (Stanford)

5 The problem with loops P(c) 0.5 Cloudy P(r) c c Rain Sprinkler P(s) c c Grass-wet deterministic or The grass is dry only if no rain and no sprinklers. 5 P(g) = P(r, s) ~ 0 Jack Breese (Microsoft) & Daphne Koller (Stanford)

6 The problem with loops contd. P(g) = 0 0 P(g r, s) P(r, s) + P(g r, s) P(r, s) + P(g r, s) P(r, s) + P(g r, s) P(r, s) 0 1 Propagation = P(r, s) ~ 0 = P(r) P(s) ~ = problem Jack Breese (Microsoft) & Daphne Koller (Stanford)

7 Variable elimination A B P(c) = S P(c b) S P(b a) P(a) b a P(A) P(B A) P(b) C Π P(B, A) S A P(B) P(C B) Π P(C, B) S B P(C) 7 Jack Breese (Microsoft) & Daphne Koller (Stanford)

8 Inference as variable elimination A factor over X is a function from val(x) to numbers in [0,1]: A CPT is a factor A joint distribution is also a factor BN inference: factors are multiplied to give new ones variables in factors summed out A variable can be summed out as soon as all factors mentioning it have been multiplied. 8 Jack Breese (Microsoft) & Daphne Koller (Stanford)

9 Variable Elimination with loops Age Exposure to Toxics Gender Smoking P(A) P(G) P(S A,G) Π P(A,G,S) S A S G P(A,S) P(E A) Π P(E,S) P(C E,S) P(A,E,S) Serum Calcium Cancer Lung Tumor Π P(E,S,C) S E,S P(C) P(L C) Π P(C,L) S C P(L) 9 Complexity is exponential in the size of the factors Jack Breese (Microsoft) & Daphne Koller (Stanford)

10 Join trees* A join tree is a partially precompiled factorization Age Gender P(A) x P(G) x P(S A,G) x A, G, S P(A,S) Exposure to Toxics Smoking A, E, S Cancer E, S, C Serum Calcium 10 Lung Tumor C, S-C * aka Junction Tree, Lauritzen-Spiegelhalter, or Hugin algorithm, C, L Jack Breese (Microsoft) & Daphne Koller (Stanford)

11 Background: Markov networks Random variable: B,E,A,J,M Joint distribution: Pr(B,E,A,J,M) x x x x Undirected graphical models give another way of defining a compact model of the joint distribution via potential functions. ϕ(a=a,j=j) is a scalar measuring the compatibility of A=a J=j A J ϕ(a,j) F F 20 F T 1 T F 0.1 T T 0.4

12 Background Pr(B = b, E = e, A = a, j, m) = 1 Z φ JA(a, j)φ MA (a, m)φ AB (a, b)φ AE (a, e)φ E (e)φ B (b) x x x x clique potential ϕ(a=a,j=j) is a scalar measuring the compatibility of A=a J=j A J ϕ(a,j) F F 20 F T 1 T F 0.1 T T 0.4

13 Another example Undirected graphical models Smoking Cancer [h/t Pedro Domingos] x = vector Asthma Cough 1 P ( x) = ÕFc( x c ) Z Z ( ) c = åõf x c c x c x c = short vector Smoking Cancer Ф(S,C) False False 4.5 False True 4.5 True False 2.7 True True 4.5 H/T: Pedro Domingos

14 Markov Networks = Markov Random Fields Undirected Graphical Model A B C

15 Markov Random Fields Undirected Graphical Model AB B BC Clique Separator Clique P(U) = Õ P(Clique) / Õ P(Separator) P(A,B,C) = P(A,B) P(B,C) / P(B)

16 Markov Random Fields A node is conditionally independent of all others given its neighbours.

17 Example Factor Graphs Exponential (joint) parameterization Pairwise parameterization A A V AB A V AB B C B C B C Markov network V ABC V AB Factor graph for joint parameterization Factor graph for pairwise parameterization

18 Transforming MRFs into BNs and back 18

19 Factor Graphs vs. MRFs 19

20 BNs MRFs FGs 20

21 Generative vs. Discriminative 21

22 Conditional Random Field A Conditional random field (CRF) is a Markov random field of unobservables which are globally conditioned on a set of observables (Lafferty et al., 2001) Lafferty, J., McCallum, A., Pereira, F. "Conditional random fields: Probabilistic models for segmenting and labeling sequence data". Proc. 18th International Conf. on Machine Learning. Morgan Kaufmann. pp

23 P(Y X)

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25 Augmenting Probabilistic Graphical Models with Ontology Information: Object Classification R. Möller Institute of Information Systems University of Luebeck

26 Based on ECCV14 paper: Large-Scale Object Recognition using Label Relation Graphs Jia Deng 1,2, Nan Ding 2, Yangqing Jia 2, Andrea Frome 2, Kevin Murphy 2, Samy Bengio 2, Yuan Li 2, Hartmut Neven 2, Hartwig Adam 2 University of Michigan 1, Google 2

27 Object Classification Assign semantic labels to objects Dog Corgi Puppy Cat

28 Object Classification Assign semantic labels to objects Probabilities Dog 0.9 Corgi Puppy Cat

29 Object Classification Assign semantic labels to objects Features Feature Extractor Classifier Probabilities Dog 0.9 Corgi Puppy Cat

30 Object Classification Independent binary classifiers: Logistic Regression Multiclass classifier: Softmax Dog Corgi Puppy Cat No assumptions about relations. + / / / / Dog Corgi Puppy Cat Assumes mutual exclusive labels.

31 Object labels have rich relations Hierarchical Exclusion Dog Dog Cat Corg i Puppy Cat Corgi Puppy Softmax: all labels are mutually exclusive L Logistic Regression: all labels overlap L Overlap

32 Goal: A new classification model Respects real world label relations Dog Cat Dog 0.9 Corgi Puppy Corgi Puppy Cat

33 Visual Model + Knowledge Graph Visual Model Joint Inference Knowledge Graph Dog 0.9 Corgi Puppy Cat Assumption in this work: Knowledge graph is given and fixed.

34 Agenda Encoding prior knowledge (HEX graph) Classification model Efficient Exact Inference

35 Agenda Encoding prior knowledge (HEX graph) Classification model Efficient Exact Inference

36 Hierarchy and Exclusion (HEX) Graph Hierarchical Exclusion Dog Cat Corgi Puppy Hierarchical edges (directed) Exclusion edges (undirected)

37 Examples of HEX graphs Dog Cat Red Shiny Person Car Bird Round Thick Male Female Child Boy Girl Mutually exclusive All overlapping Combination

38 State Space: Legal label configurations Each edge defines a constraint. Corgi Dog Puppy Cat Dog Cat Corgi Puppy

39 State Space: Legal label configurations Each edge defines a constraint. Dog Cat Corgi Puppy Hierarchy: (dog, corgi) can t be (0,1) Dog Cat Corgi Puppy

40 State Space: Legal label configurations Each edge defines a constraint. Dog Cat Corgi Puppy Hierarchy: (dog, corgi) can t be (0,1) Exclusion: (dog, cat) can t be (1,1) Dog Cat Corgi Puppy

41 Agenda Encoding prior knowledge (HEX graph) Classification model Efficient Exact Inference

42 HEX Classification Model Pairwise Conditional Random Field (CRF) x R n Input scores y {0,1} n Binary Label vector Pr(y x) = 1 Z(x) i φ i (x i, y i ) i, j ψ i, j (y i, y j )

43 HEX Classification Model Pairwise Conditional Random Field (CRF) x R n Input scores y {0,1} n Binary Label vector Pr(y x) = 1 Z(x) i φ i (x i, y i ) i, j ψ i, j (y i, y j ) φ i (x i, y i ) = sigmoid(x i ) 1 sigmoid(x i ) if y i =1 if y i = 0 Unary: same as logistic regression

44 HEX Classification Model Pairwise Conditional Random Field (CRF) x R n Input scores y {0,1} n Binary Label vector Pr(y x) = 1 Z(x) i φ i (x i, y i ) i, j ψ i, j (y i, y j ) φ i (x i, y i ) = sigmoid(x i) 1 sigmoid(x i ) if y i =1 if y i = 0 Unary: same as logistic regression ψ i, j (y i, y j ) = 0 1 If violates constraints Otherwise Pairwise: set illegal configuration to zero

45 HEX Classification Model Pairwise Conditional Random Field (CRF) x R n Input scores y {0,1} n Binary Label vector Pr(y x) = 1 Z(x) φ i (x i, y i ) ψ i, j (y i, y j ) i i, j Z(x) = φ i (x i, y i ) ψ i, j (y i, y j ) y {0,1} n i Partition function: Sum over all (legal) configurations i, j

46 HEX Classification Model Pairwise Conditional Random Field (CRF) x R n Input scores y {0,1} n Binary Label vector Pr(y x) = 1 Z(x) Pr(y i =1 x) = 1 Z(x) i φ i (x i, y i ) i, j ψ i, j (y i, y j ) Probability of a single label: marginalize allother labels. y:y i =1 φ i (x i, y i ) ψ i, j (y i, y j ) i i, j

47 Special Case of HEX Model Softmax Logistic Regressions Dog Cat Red Shiny Car Bird Round Thick Mutually exclusive All overlapping Pr(y i =1 x) = exp(x i ) j 1+ exp(x j ) Pr(y i =1 x) = 1 1+ exp( x i )

48 Learning Dog Corgi 1? Puppy? Cat? Loss = logpr(dog = 1) Label: Dog Dog Pr(Dog = 1) Cat Puppy DNN Corgi Back Propagation Maximize marginal probability of observed labels DNN = Deep Neural Network

49 Agenda Encoding prior knowledge (HEX graph) Classification model Efficient Exact Inference

50 Naïve Exact Inference is Intractable Inference: Computing partition function Perform marginalization HEX-CRF can be densely connected (large treewidth)

51 Observation 1: Exclusions are good Dog Cat Car Bird Number of legal states is O(n), not O(2 n ). Lots of exclusions à Small state space à Efficient inference Realistic graphs have lots of exclusions. Rigorous analysis in paper.

52 Observation 2: Equivalent graphs Dog Cat Dog Cat Corgi Corgi Cardigan Welsh Corgi Pembroke Welsh Corgi Puppy Cardigan Welsh Corgi Pembroke Welsh Corgi Puppy

53 Observation 2: Equivalent graphs Dog Cat Dog Cat Dog Cat Corgi Corgi Corgi Cardigan Welsh Corgi Pembroke Welsh Corgi Puppy Cardigan Welsh Corgi Pembroke Welsh Corgi Puppy Cardigan Welsh Corgi Pembroke Welsh Corgi Puppy Sparse equivalent Small Treewidth J Dynamic programming Dense equivalent Prune states J Can brute force

54 A B F B E D C G F B C F HEX Graph Inference A B E D C G F A B E D C G F A B E D C G F A B F B E D C G F B C F 2.Build Junction Tree (offline)