infinite dynamic bayesian networks
DESCRIPTION
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 - PowerPoint PPT PresentationTRANSCRIPT
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