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Infinite dynamic bayesian networks

Presented by Patrick Dallaire – DAMAS Workshop november 2 th 2012. Infinite dynamic bayesian networks. ( Doshi et al. 2011). INTRODUCTION. PROBLEM DESCRIPTION. Consider precipitations measured by 500 different weather stations in USA. Observations were discretized into 7 values

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Infinite dynamic bayesian networks

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  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

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