Probabilistic Inference Lecture 1

1. Probabilistic InferenceLecture 1 M. Pawan Kumar pawan.kumar@ecp.fr Slides available online http://cvc.centrale-ponts.fr/personnel/pawan/

2. About the Course • 7 lectures + 1 exam • Probabilistic Models – 1 lecture • Energy Minimization – 4 lectures • Computing Marginals – 2 lectures • Related Courses • Probabilistic Graphical Models (MVA) • Structured Prediction

3. Instructor • Assistant Professor (2012 – Present) • Center for Visual Computing • 12 Full-time Faculty Members • 2 Associate Faculty Members • Research Interests • Probabilistic Models • Machine Learning • Computer Vision • Medical Image Analysis

4. Students • Third year at ECP • Specializing in Machine Learning and Vision • Prerequisites • Probability Theory • Continuous Optimization • Discrete Optimization

5. Outline • Probabilistic Models • Conversions • Exponential Family • Inference Example (on board) !!

6. Outline • Probabilistic Models • Markov Random Fields (MRF) • Bayesian Networks • Factor Graphs • Conversions • Exponential Family • Inference

7. MRF Unobserved Random Variables Neighbors Edges define a neighborhood over random variables

8. MRF V1 V2 V3 V4 V5 V6 V7 V8 V9 = {l1, l2,…, lh} Variable Va takes a value or a label va from a set L V = v is called a labeling Discrete, Finite

9. MRF V1 V2 V3 V4 V5 V6 V7 V8 V9 MRF assumes the Markovian property for P(v)

10. MRF V1 V2 V3 V4 V5 V6 V7 V8 V9 Va is conditionally independent of Vb given Va’s neighbors Hammersley-Clifford Theorem

11. MRF Potential ψ12(v1,v2) V1 V2 V3 Potential ψ56(v5,v6) V4 V5 V6 V7 V8 V9 Probability P(v) can be decomposed into clique potentials

12. MRF Potential ψ1(v1,d1) d1 d2 d3 V1 V2 V3 Observed Data d4 d5 d6 V4 V5 V6 d7 d8 d9 V7 V8 V9 Probability P(v) proportional to Π(a,b)ψab(va,vb) Probability P(d|v) proportional to Πaψa(va,da)

13. MRF d1 d2 d3 V1 V2 V3 d4 d5 d6 V4 V5 V6 d7 d8 d9 V7 V8 V9 Πaψa(va,da) Π(a,b)ψab(va,vb) Probability P(v,d) = Z Z is known as the partition function

14. MRF d1 d2 d3 V1 V2 V3 d4 d5 d6 V4 V5 V6 High-order Potential ψ4578(v4,v5,v7,v8) d7 d8 d9 V7 V8 V9

15. Pairwise MRF Unary Potential ψ1(v1,d1) d1 d2 d3 V1 V2 V3 d4 d5 d6 Pairwise Potential ψ56(v5,v6) V4 V5 V6 d7 d8 d9 V7 V8 V9 Πaψa(va,da) Π(a,b)ψab(va,vb) Probability P(v,d) = Z Z is known as the partition function

16. MRF d1 d2 d3 V1 V2 V3 d4 d5 d6 V4 V5 V6 d7 d8 d9 V7 V8 V9 A is conditionally independent of B given C if there is no path from A to B when C is removed

17. Conditional Random Fields (CRF) d1 d2 d3 V1 V2 V3 d4 d5 d6 V4 V5 V6 d7 d8 d9 V7 V8 V9 CRF assumes the Markovian property for P(v|d) Hammersley-Clifford Theorem

18. CRF d1 d2 d3 V1 V2 V3 d4 d5 d6 V4 V5 V6 d7 d8 d9 V7 V8 V9 Probability P(v|d) proportional to Πaψa(va;d) Π(a,b)ψab(va,vb;d) Clique potentials that depend on the data

19. CRF d1 d2 d3 V1 V2 V3 d4 d5 d6 V4 V5 V6 d7 d8 d9 V7 V8 V9 Πaψa(va;d)Π(a,b)ψab(va,vb;d) Probability P(v|d) = Z Z is known as the partition function

20. MRF and CRF V1 V2 V3 V4 V5 V6 V7 V8 V9 Πaψa(va) Π(a,b)ψab(va,vb) Probability P(v) = Z

21. Outline • Probabilistic Models • Markov Random Fields (MRF) • Bayesian Networks • Factor Graphs • Conversions • Exponential Family • Inference

22. Bayesian Networks V1 V2 V3 V4 V5 V6 V7 V8 Directed Acyclic Graph (DAG) – no directed loops Ignoring directionality of edges, a DAG can have loops

23. Bayesian Networks V1 V2 V3 V4 V5 V6 V7 V8 Bayesian Network concisely represents the probability P(v)

24. Bayesian Networks V1 V2 V3 V4 V5 V6 V7 V8 Probability P(v) = Πa P(va|Parents(va)) P(v1)P(v2|v1)P(v3|v1)P(v4|v2)P(v5|v2,v3)P(v6|v3)P(v7|v4,v5)P(v8|v5,v6)

25. Bayesian Networks Courtesy Kevin Murphy

26. Bayesian Networks V1 V2 V3 V4 V5 V6 V7 V8 Va is conditionally independent of its ancestors given its parents

27. Bayesian Networks Conditional independence of A and B given C Courtesy Kevin Murphy

28. Outline • Probabilistic Models • Markov Random Fields (MRF) • Bayesian Networks • Factor Graphs • Conversions • Exponential Family • Inference

29. Factor Graphs V1 V2 V3 a b c d e V4 V5 V6 f g Two types of nodes: variable nodes and factor nodes Bipartite graph between the two types of nodes

30. Factor Graphs ψa(v1,v2) V1 V2 V3 a b c d e V4 V5 V6 f g Factor graphs concisely represents the probability P(v)

31. Factor Graphs ψa({v}a) V1 V2 V3 a b c d e V4 V5 V6 f g Factor graphs concisely represents the probability P(v)

32. Factor Graphs ψb(v2,v3) V1 V2 V3 a b c d e V4 V5 V6 f g Factor graphs concisely represents the probability P(v)

33. Factor Graphs ψb({v}b) V1 V2 V3 a b c d e V4 V5 V6 f g Factor graphs concisely represents the probability P(v)

34. Factor Graphs ψb({v}b) V1 V2 V3 a b c d e V4 V5 V6 f g Πaψa({v}a) Probability P(v) = Z Z is known as the partition function

35. Outline • Probabilistic Models • Conversions • Exponential Family • Inference

36. MRF to Factor Graphs

37. Bayesian Networks to Factor Graphs

38. Factor Graphs to MRF

39. Outline • Probabilistic Models • Conversions • Exponential Family • Inference

40. Motivation Random Variable V Label set L = {l1, l2,…, lh} Samples V1, V2, …, Vm that are i.i.d. Functions ϕα: LReals α indexes a set of functions Empirical expectations: μα = (Σiϕα(Vi))/m Expectation wrt distribution P: EP[ϕα(V)] = Σiϕα(li)P(li) Given empirical expectations, find compatible distribution Underdetermined problem

41. Maximum Entropy Principle max Entropy of the distribution s.t. Distribution is compatible

42. Maximum Entropy Principle max -Σi P(li)log(P(li)) s.t. Distribution is compatible

43. Maximum Entropy Principle max -Σi P(li)log(P(li)) s.t.Σiϕα(li)P(li) = μα for all α Σi P(li) = 1 P(v) proportional to exp(-Σαθαϕα(v))

44. Exponential Family Random Variable V = {V1, V2, …,Vn} Label set L = {l1, l2,…, lh} Labeling V = v, va L for all a  {1, 2,…, n} Functions ϕα: LnReals α indexes a set of functions P(v) = exp{-ΣαθαΦα(v) - A(θ)} Sufficient Statistics Normalization Constant Parameters

45. Minimal Representation P(v) = exp{-ΣαθαΦα(v) - A(θ)} Sufficient Statistics Normalization Constant Parameters No non-zero c such that Σα cαΦα(v) = Constant

46. Ising Model P(v) = exp{-ΣαθαΦα(v) - A(θ)} Random Variable V = {V1, V2, …,Vn} Label set L = {l1, l2}

47. Ising Model P(v) = exp{-ΣαθαΦα(v) - A(θ)} Random Variable V = {V1, V2, …,Vn} Label set L = {-1, +1} Neighborhood over variables specified by edges E Sufficient Statistics Parameters for all Va V va θa vavb θab for all (Va,Vb)  E

48. Ising Model P(v) = exp{-Σaθava -Σa,bθabvavb- A(θ)} Random Variable V = {V1, V2, …,Vn} Label set L = {-1, +1} Neighborhood over variables specified by edges E Sufficient Statistics Parameters for all Va V va θa vavb θab for all (Va,Vb)  E

49. Interactive Binary Segmentation

50. Interactive Binary Segmentation Foreground histogram of RGB values FG Background histogram of RGB values BG ‘+1’ indicates foreground and ‘-1’ indicates background