Infinite dynamic bayesian networks

1. Presented by Patrick Dallaire – DAMAS Workshop november 2th 2012 Infinite dynamic bayesian networks (Doshi et al. 2011)

2. INTRODUCTION

3. PROBLEM DESCRIPTION • Consider precipitations measured by 500 different weather stations in USA. • Observations were discretized into 7 values • The dataset consists of a time series including 3,287 daily measures • How can we learn the underlying weather model that produced the sequence of precipitations?

4. HIDDEN MARKOV MODEL • Observations are produced based on the hidden state • The hidden state evolvesaccording to some dynamics • Markov assumption says that summarizes the states history and is thus enough to generate • The learning task is to infer and from data

5. INFINITE DYNAMIC BAYESIAN NETWORKS

6. TRANSITION MODEL • A regular DBN is a directed graphical model • State at time is represented through a set of factors

7. TRANSITION MODEL • A regular DBN is a directed graphical model • State at time is represented through a set of factors • The next state is sampledaccording to:where representsthe values of the parents

8. TRANSITION MODEL • A regular DBN is a directed graphical model • State at time is represented through a set of factors • The next state is sampledaccording to:where representsthe values of the parents

9. OBSERVATION MODEL • The state of a DBN isgenerally hidden • State values must be inferred from a set of observable nodes • The observations are sampled from:where is the values of the parents

10. OBSERVATION MODEL • The state of a DBN isgenerally hidden • State values must be inferred from a set of observable nodes • The observations are sampled from:where is the values of the parents

11. OBSERVATION MODEL • The state of a DBN isgenerally hidden • State values must be inferred from a set of observable nodes • The observations are sampled from:where is the values of the parents

12. LEARNING THE STRUCTURE • The number of hidden factors is unknown • The state transition structure is unknown • The state observation structure is unknown

13. PRIOR OVER DBN STRUCTURES • A Bayesian nonparametric prior is specified over structures with Indian buffet processes (IBP) • We specify a prior over observation connection structures: • We specify a prior over transition connection structures:

14. IBP ON OBSERVATION STRUCTURE

15. IBP ON OBSERVATION STRUCTURE

16. IBP ON OBSERVATION STRUCTURE • selects a parent factor with probability • samplesnew parent factors

17. IBP ON OBSERVATION STRUCTURE • selects a parent factor with probability • samplesnew parent factors

18. IBP ON OBSERVATION STRUCTURE • selects a parent factor with probability • samplesnew parent factors

19. IBP ON OBSERVATION STRUCTURE • selects a parent factor with probability • samplesnew parent factors

20. IBP ON OBSERVATION STRUCTURE • selects a parent factor with probability • samplesnew parent factors

21. IBP ON OBSERVATION STRUCTURE • selects a parent factor with probability • samplesnew parent factors

22. IBP ON OBSERVATION STRUCTURE • selects a parent factor with probability • samplesnew parent factors

23. IBP ON TRANSITION STRUCTURE

24. IBP ON TRANSITION STRUCTURE

25. IBP ON TRANSITION STRUCTURE

26. IBP ON TRANSITION STRUCTURE • selects a parent factor with probability • samplesnew parent factors

27. IBP ON TRANSITION STRUCTURE • selects a parent factor with probability • samplesnew parent factors

28. IBP ON TRANSITION STRUCTURE • selects a parent factor with probability • samplesnew parent factors

29. IBP ON TRANSITION STRUCTURE • selects a parent factor with probability • samplesnew parent factors

30. IBP ON TRANSITION STRUCTURE • selects a parent factor with probability • samplesnew parent factors

31. IBP ON TRANSITION STRUCTURE • selects a parent factor with probability • samplesnew parent factors

32. IBP ON TRANSITION STRUCTURE • selects a parent factor with probability • samplesnew parent factors

33. GRAPHICAL MODEL OF THE PRIOR

34. LEARNING MODEL DISTRIBUTIONS • The observation distribution is unknown • The transition distribution is unknown

35. PRIOR OVER DBN DISTRIBUTIONS • A Bayesian prior is specified over observation distributions: where is a prior base distribution

36. PRIOR OVER DBN DISTRIBUTIONS • A Bayesian prior is specified over observation distributions: where is a prior base distribution • A Bayesian nonparametric prior is specified over transition distributions:where is a Dirichlet process and is a Stickbreaking distribution

37. PRIOR ON OBSERVATION MODEL • For each observable variable , we can draw an observation distribution from:

38. PRIOR ON OBSERVATION MODEL • For each observable variable , we can draw an observation distribution from: • Assuming is discrete, could be a Dirichlet

39. PRIOR ON OBSERVATION MODEL • For each observable variable , we can draw an observation distribution from: • Assuming is discrete, could be a Dirichlet • The prior could also be a Dirichlet

40. PRIOR ON OBSERVATION MODEL • For each observable variable , we can draw an observation distribution from: • Assuming is discrete, could be a Dirichlet • The prior could also be a Dirichlet • The posterior is obtained by counting how many times specific observations occurred

41. EXAMPLE

42. EXAMPLE

43. EXAMPLE red blue

44. EXAMPLE red blue

45. PRIOR ON TRANSITION MODEL • First, we sample the expected factor transition distribution:

46. PRIOR ON TRANSITION MODEL • First, we sample the expected factor transition distribution: • For each active hidden factor, we sample an individual transition distribution: where controls the variance around

47. PRIOR ON TRANSITION MODEL • First, we sample the expected factor transition distribution for infinitely many factors: • For each active hidden factor, we sample an individual transition distribution: where controls the (inverse) variance • The posterior is again obtained by counting

48. EXAMPLE

49. EXAMPLE

50. EXAMPLE