infinite dynamic bayesian networks

INFINITE DYNAMIC BAYESIAN

NETWORKS

Presented by Patrick Dallaire – DAMAS Workshop november 2th 2012

(Doshi et al. 2011)

INTRODUCTION

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?

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

INFINITE DYNAMIC BAYESIAN NETWORKS

TRANSITION MODEL• A regular DBN is a directed graphical

model• State at time is represented through a

set of factors

TRANSITION MODEL• A regular DBN is a directed graphical

model• State at time is represented through a

set of factors • The next state is sampled

according to:

where representsthe values of the parents

TRANSITION MODEL• A regular DBN is a directed graphical

model• State at time is represented through a

set of factors • The next state is sampled

according to:

where representsthe values of the parents

OBSERVATION MODEL• The state of a DBN is

generally hidden• State values must be

inferred from a set of observable nodes

• The observations are sampled from:

where is the values of the parents

OBSERVATION MODEL• The state of a DBN is

generally hidden• State values must be

inferred from a set of observable nodes

• The observations are sampled from:

where is the values of the parents

OBSERVATION MODEL• The state of a DBN is

generally hidden• State values must be

inferred from a set of observable nodes

• The observations are sampled from:

where is the values of the parents

LEARNING THE STRUCTURE• The number of hidden factors is unknown

• The state transition structure is unknown

• The state observation structure is unknown

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:

IBP ON OBSERVATION STRUCTURE

IBP ON OBSERVATION STRUCTURE

1) selects a parent factor with probability

2) samplesnew parent factors

IBP ON OBSERVATION STRUCTURE

1) selects a parent factor with probability

2) samplesnew parent factors

IBP ON OBSERVATION STRUCTURE

1) selects a parent factor with probability

2) samplesnew parent factors

IBP ON OBSERVATION STRUCTURE

1) selects a parent factor with probability

2) samplesnew parent factors

IBP ON OBSERVATION STRUCTURE

1) selects a parent factor with probability

2) samplesnew parent factors

IBP ON OBSERVATION STRUCTURE

1) selects a parent factor with probability

2) samplesnew parent factors

IBP ON OBSERVATION STRUCTURE

1) selects a parent factor with probability

2) samplesnew parent factors

IBP ON OBSERVATION STRUCTURE

IBP ON TRANSITION STRUCTURE

IBP ON TRANSITION STRUCTURE

IBP ON TRANSITION STRUCTURE

IBP ON TRANSITION STRUCTURE

1) selects a parent factor with probability

2) samplesnew parent factors

IBP ON TRANSITION STRUCTURE

1) selects a parent factor with probability

2) samplesnew parent factors

IBP ON TRANSITION STRUCTURE

1) selects a parent factor with probability

2) samplesnew parent factors

IBP ON TRANSITION STRUCTURE

1) selects a parent factor with probability

2) samplesnew parent factors

IBP ON TRANSITION STRUCTURE

1) selects a parent factor with probability

2) samplesnew parent factors

IBP ON TRANSITION STRUCTURE

1) selects a parent factor with probability

2) samplesnew parent factors

IBP ON TRANSITION STRUCTURE

1) selects a parent factor with probability

2) samplesnew parent factors

GRAPHICAL MODEL OF THE PRIOR

LEARNING MODEL DISTRIBUTIONS

• The observation distribution is unknown

• The transition distribution is unknown

PRIOR OVER DBN DISTRIBUTIONS• A Bayesian prior is specified over

observation distributions:

where is a prior base distribution

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

PRIOR ON OBSERVATION MODEL• For each observable variable , we can

draw an observation distribution from:

PRIOR ON OBSERVATION MODEL• For each observable variable , we can

draw an observation distribution from:

• Assuming is discrete, could be a Dirichlet

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

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

EXAMPLE

EXAMPLE

EXAMPLE

red

blue

EXAMPLE

red

blue

PRIOR ON TRANSITION MODEL• First, we sample the expected factor

transition distribution:

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

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

EXAMPLE

EXAMPLE

EXAMPLE

EXAMPLE

EXAMPLE

EXAMPLE

EXAMPLE

EXAMPLE

EXAMPLE

GRAPHICAL MODEL OF THE PRIOR

GRAPHICAL MODEL OF THE PRIOR

PRIOR SUMMARY• States are represented by infinitely many

factors by using a recursive IBP prior

• Factors can take infinitely many values by using a Hierarchical Dirichlet process prior

• Only a finite number of factors are used to explain the observations with probability 1

INFERENCE

factor/factor connections Gibbs sampling

factor/observation connections

Gibbs sampling

transitions Dirichlet-multinomial

observations Dirichlet-multinomial

state sequence Factored frontier algorithm

Add/delete factors M-H birth/death

DOSHI’S RESULTS

DOSHI’S RESULTS

APPLYING ECIBP TO IDBN

OBSERVATION MODEL EXTENSION• We modify the Indian buffet process prior

on factor to observation connections

• We propose the extended cascading Indian buffet process on hidden factors’ structure to explain observations

• This would extend the iDBN model to consider structure among factors of the same time slice

eCIBP ON OBSERVATION STRUCTURE

eCIBP ON OBSERVATION STRUCTURE

eCIBP ON OBSERVATION STRUCTURE

eCIBP ON OBSERVATION STRUCTURE

eCIBP ON OBSERVATION STRUCTURE

eCIBP ON OBSERVATION STRUCTURE

eCIBP ON OBSERVATION STRUCTURE

eCIBP ON OBSERVATION STRUCTURE

eCIBP ON OBSERVATION STRUCTURE

eCIBP ON OBSERVATION STRUCTURE• The previous sequence

was the cascading Indian buffet process

eCIBP ON OBSERVATION STRUCTURE• The previous sequence

was the cascading Indian buffet process

• The extended CIBP samples connections that jump over layers

eCIBP ON OBSERVATION STRUCTURE• The previous sequence

was the cascading Indian buffet process

• The extended CIBP samples connections that jump over layers

eCIBP ON OBSERVATION STRUCTURE• The previous sequence

was the cascading Indian buffet process

• The extended CIBP samples connections that jump over layers

eCIBP ON OBSERVATION STRUCTURE• The previous sequence

was the cascading Indian buffet process

• The extended CIBP samples connections that jump over layers

eCIBP ON OBSERVATION STRUCTURE• The previous sequence

was the cascading Indian buffet process

• The extended CIBP samples connections that jump over layers

Not allowed

iDBN with recursive IBP iDBN with eCIBP• Dependency among

factors of the same time slice are not allowed

• Hierarchical layered structure is achieved with higher order Markov models

• Uses recursive IBP + IBP• Can model a subset of

all possible DBN structures

• Structure among factors in the same time slice can be any DAG

• DAG structure is achieved with first order Markov models

• Uses recursive IBP + eCIBP

• Can model all possible DBN structures

CONCLUSION