Variational Methods for Graphical Models

1. Variational Methods for Graphical Models Micheal I. Jordan Zoubin Ghahramani Tommi S. Jaakkola Lawrence K. Saul Presented by: Afsaneh Shirazi

2. Outline • Motivation • Inference in graphical models • Exact inference is intractable • Variational methodology • Sequential approach • Block approach • Conclusions

3. Motivation(Example: Medical Diagnosis) diseases What is the most probable disease? symptoms

4. Motivation • We want to answer some queries about our data • Graphical model is a way to model data • Inference in some graphical models is intractable (NP-hard) • Variational methods simplify the inference in graphical models by using approximation

5. Graphical Models • Directed (Bayesian network) • Undirected P(S3|S1,S2) P(S1) S1 S4 P(S4|S3) S3 P(S2) S2 S5 P(S5|S3,S4) (C1) (C3) (C2)

6. Inference in Graphical Models Inference:Given a graphical model, the process of computing answers to queries • How computationally hard is this decision problem? • Theorem:Computing P(X = x) in a Bayesian network is NP-hard

7. Why Exact Inference is Intractable? diseases Diagnose the most probable disease symptoms

8. Why Exact Inference is Intractable? diseases : Observed symptoms symptoms

9. Why Exact Inference is Intractable? diseases 1 0 1 :Noisy-OR model symptoms

10. Why Exact Inference is Intractable? diseases 1 0 1 : Noisy-OR model symptoms

11. Why Exact Inference is Intractable?

12. Why Exact Inference is Intractable? diseases : Observed symptoms symptoms

13. Why Exact Inference is Intractable? diseases : Observed symptoms symptoms

14. Reducing the Computational Complexity Simple graph for exact methods Variational Methods Approximate the probability distribution Use the role of convexity

15. Express a Function Variationally • is a concave function

16. Express a Function Variationally • is a concave function

17. Express a Function Variationally • If the function is not convex or concave: transform the function to a desired form • Example: logistic function Transforming back Transformation Approximation

18. Approaches to Variational Methods • Sequential Approach: (on-line) nodes are transformed in an order, determined during inference process • Block Approach: (off-line) has obvious substructures

19. Completely transformed Graph Reintroduce one node at a time Simple Graph for exact methods Sequential Approach(Two Methods) Simple Graph for exact methods Untransformed Graph Transform one node at a time

20. Sequential Approach (Example) diseases Log Concave symptoms

21. Sequential Approach (Example) diseases Log Concave symptoms

22. Sequential Approach (Example) diseases 1 symptoms

23. Sequential Approach (Example) diseases 1 symptoms

24. Sequential Approach (Example) diseases 1 symptoms

25. Sequential Approach (Upper Bound and Lower Bound) • We need both lower bound and upper bound

26. How to Compute Lower Bound for a Concave Function? • Lower bound for concave functions: Variational parameter is probability distribution

27. Block Approach (Overview) • Off-line application of sequential approach • Identify some structure amenable to exact inference • Family of probability distribution via introduction of parameters • Choose best approximation based on evidence

28. Minimize KL divergence Family of Block Approach (Details) • KL divergence

29. Block Approach (Example – Boltzmann machine) Si Sj

30. Block Approach (Example – Boltzmann machine) Si Sj=1

31. Block Approach (Example – Boltzmann machine) si sj

32. Block Approach (Example – Boltzmann machine) Minimize KL Divergence si sj

33. Block Approach (Example – Boltzmann machine) Minimize KL Divergence si sj Mean field equations: solve for fixed point

34. Conclusions • Time or space complexity of exact calculation is unacceptable • Complex graphs can be probabilistically simple • Inference in simplified models provides bounds on probabilities in the original model

35. Thank You

36. Extra Slides

37. Concerns • Approximation accuracy • Strong dependencies can be identified • Not based on convexity transformation • Not able to assure that the framework will transfer to other examples • Not straightforward to develop a variational approximation for new architectures

38. Justification for KL Divergence • Best lower bound on the probability of the evidence

39. KL Divergence between Q(H|E) and P(H|E,) EM • Maximum likelihood parameter estimation: • Following function is the lower bound on log likelihood

40. Traditional EM EM • Maximize the bound with respect to Q • Fix Q, maximize with respect to Approximation to EM algorithm

41. DAG Junction Tree Initialization Inconsistent Junction Tree Propagation Consistent Junction Tree Marginalization Principle of Inference

42. X1 X2 Y1 Y2 X1,Y1 X2,Y2 X1,X2 X1 X2 Example: Create Join Tree HMM with 2 time steps: Junction Tree:

43. X1,Y1 X2,Y2 X1,X2 X1 X2 Example: Initialization

44. Example: Collect Evidence • Choose arbitrary clique, e.g. X1,X2, where all potential functions will be collected. • Call recursively neighboring cliques for messages: • 1. Call X1,Y1. • 1. Projection: • 2. Absorption:

45. X1,Y1 X2,Y2 X1,X2 X1 X2 Example: Collect Evidence (cont.) • 2. Call X2,Y2: • 1. Projection: • 2. Absorption:

46. Example: Distribute Evidence • Pass messages recursively to neighboring nodes • Pass message from X1,X2 to X1,Y1: • 1. Projection: • 2. Absorption:

47. X1,Y1 X2,Y2 X1,X2 X1 X2 Example: Distribute Evidence (cont.) • Pass message from X1,X2 to X2,Y2: • 1. Projection: • 2. Absorption:

48. Example: Inference with evidence • Assume we want to compute: P(X2|Y1=0,Y2=1) (state estimation) • Assign likelihoods to the potential functions during initialization:

49. Example: Inference with evidence (cont.) • Repeating the same steps as in the previous case, we obtain:

50. Variable Elimination General idea: • Write query in the form • Iteratively • Move all irrelevant terms outside of innermost sum • Perform innermost sum, getting a new term • Insert the new term into the product