Learning Hierarchical Bayesian Networks for Large-Scale Data Analysis

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Learning Hierarchical Bayesian Networks
for Large-Scale Data Analysis
Kyu-Baek Hwang
1
,Byoung-Hee Kim
2
,and Byoung-Tak Zhang
2
1
School of Computing,Soongsil University,
Seoul 156-743,Korea
kbhwang@ssu.ac.kr
2
School of Computer Science and Engineering
Seoul National University,
Seoul 151-742,Korea
bhkim@bi.snu.ac.kr
,
btzhang@cse.snu.ac.kr
Abstract.
Bayesian network learning is a useful tool for exploratory
data analysis.However,applying Bayesian networks to the analysis of
large-scale data,consisting of thousands of attributes,is not straight-
forward because of the heavy computational burden in learning and vi-
sualization.In this paper,we propose a novel method for large-scale
data analysis based on hierarchical compression of information and con-
strained structural learning,i.e.,hierarchical Bayesian networks (HBNs).
The HBN can compactly visualize global probabilistic structure through
a small number of hidden variables,approximately representing a large
number of observed variables.An efficient learning algorithm for HBNs,
which incrementally maximizes the lower bound of the likelihood func-
tion,is also suggested.The effectiveness of our method is demonstrated
by the experiments on synthetic large-scale Bayesian networks and a
real-life microarray dataset.
1 Introduction
Due to their ability to caricature conditional independencies among variables,
Bayesian networks have been applied to various data mining tasks [9],[4].How-
ever,application of the Bayesian network to extremely large domains (e.g.,a
database consisting of thousands of attributes) still remains a challenging task.
General approach to structural learning o
f Bayesian networks,i.e.,greedy search,
encounters the following problems when the number of variables is greater than
several thousands.First,the amount of running time for the structural learning
is formidable.Moreover,greedy search is likely to be trapped in local optima,
because of the increased search space.
Until now,several researchers suggested the methods for alleviating the above
problems [5],[8],[6].Even though these approaches have been shown to effi-
ciently find a reasonable solution,they have the following two drawbacks.First,
they are likely to spend lots of time to learn local structure,which might be
I.King et al.(Eds.):ICONIP 2006,Part I,LNCS 4232,pp.670–679,2006.
c

Springer-Verlag Berlin Heidelberg 2006
Learning Hierarchical Bayesian Netw
orks for Large-Scale Data Analysis 671
less important in the viewpoint of grasping global structure.The second prob-
lem is about visualization.It would be extremely hard to extract useful knowl-
edge from a complex network structure consisting of thousands of vertices and
edges.
In this paper,we propose a new method for large-scale data analysis using
hierarchical Bayesian networks.It should be noted that introducing hierarchical
structures in modeling is a generic technique.Several researchers have intro-
duced the hierarchy to probabilistic graphical modeling [10],[7].Our approach
is different from theirs in the purpose of hierarchical modeling.The purpose
of our method is to make it feasible to apply probabilistic graphical modeling
to extremely large domain
.We also propose an efficien
t learning algorithm for
hierarchical Bayesian networks having lots of hidden variables.
The paper is organized as follows.In S
ection 2,we define the hierarchical
Bayesian network (HBN) and describe its property.The learning algorithm for
HBNs is described in Section 3.In Sectio
n 4,we demonstrate the effectiveness
of our method through the experiments on various large-scale datasets.Finally,
we draw the conclusion in Section 5.
2 Definition of the Hierarchical Bayesian Network for
Large-Scale Data Analysis
Assume that our problem domain is described by
n
discrete variables,
Y
=
{
Y
1
,Y
2
,...,Y
n
}
.
1
The hierarchical Bayesian network for this domain is a spe-
cial Bayesian network,consisting of
Y
and additional hidden variables.It as-
sumes a layered hierarchical structure as follows.The bottom layer (observed
layer) consists of the observed variables
Y
.The first hidden layer consists of

n/
2

hidden variables,
Z
1
=
{
Z
11
,Z
12
,...,Z
1

n/
2

}
.The second hidden layer
consists of

(

n/
2

)
/
2

hidden variables,
Z
2
=
{
Z
21
,Z
22
,...,Z
2

(

n/
2

)
/
2

}
.Fi-
nally,the top layer (the

log
2
n

-th hidden layer) consists of only one hid-
den variable,
Z

log
2
n

=
{
Z

log
2
n

1
}
.We indicate all the hidden variables as
Z
=
{
Z
1
,
Z
2
,...,
Z

log
2
n

}
.
2
Hierarchical Bayesian networks,consisting of the
variables
{
Y
,
Z
}
,have the following structural constraints.
1.Any parents of a variable should be in the same or immediate upper layer.
2.At most,one parent from the immediate upper layer is allowed for each
variable.
Fig.1 shows an example HBN consisting of eight observed and seven hidden
variables.By the above structural constr
aints,a hierarchical Bayesian network
represents the joint probability distribution over
{
Y
,
Z
}
as follows.
1
In this paper,we represent a random variable as a capital letter (e.g.,
X
,
Y
,and
Z
) and a set of variables as a boldface capital letter (e.g.,
X
,
Y
,and
Z
).The
corresponding lowercase letters denote the instantiation of the variable (e.g.,
x
,
y
,
and
z
) or all the members of the set of variables (e.g.,
x
,
y
,and
z
),respectively.
2
We assume that all hidden variables are also discrete.
672 K.-B.Hwang,B.-H.Kim,and B.-T.Zhang
Z
11
Observed layer
Hidden layer 2
Hidden layer 3
Hidden layer 1
Y
2
Y
8
Y
7
Y
6
Y
5
Y
4
Y
3
Y
1
Z
12
Z
13
Z
14
Z
21
Z
22
Z
31
Fig.1.
An example HBN structure.The bottom(observed) layer consists of eight vari-
ables describing the problem domain.Each hidden layer corresponds to a compressed
representation of the observed layer.
P
(
Y
,
Z
)=
P
(
Z

log
2
n

)
·



log
2
n
−
1

i
=1
P
(
Z
i
|
Z
i
+1
)


·
P
(
Y
|
Z
1
)
.
(1)
An HBN is specified by the tuple

S
h
,
Θ
S
h

.Here,
S
h
denotes the structure of
HBNand
Θ
S
h
denotes the set of the parameters of local probability distributions
given
S
h
.In addition,we denote the parameters for each layer as
{
Θ
S
h
Y
,
Θ
S
h
Z
1
,...,
Θ
S
h
Z
￿
log
2
n
￿
}
(=
Θ
S
h
).
The hierarchical Bayesian network can r
epresent hierarchical compression of
the information contained in
Y
=
{
Y
1
,Y
2
,...,Y
n
}
.The number of hidden vari-
ables comprising the first hidden layer is half of
n
.Under the structural con-
straints,each hidden variable in the first hidden layer,i.e.,
Z
1
i
(1

i
≤
n/
2

),
can have two child nodes in the observed layer as depicted in Fig 1.
3
Here,each
hidden variable corresponds to a compressed representation of its own children
if the number of possible values of it is less than the number of possible config-
urations of its children.
In the hierarchical Bayesian network,e
dges between the variables of the same
layer are also allowed (e.g.,see hidden layers 1 and 2 in Fig.1).These edges
encode the conditional (in)dependencies among the variables in the same layer.
The conditional independencies among the variables in a hidden layer corre-
spond to a rough representation of the conditional independencies among the
observed variables because each hidden variable is a compressed representation
for a set of observed variables.When we deal with a problem domain consisting
of thousands of observed variables,an approximated probabilistic dependencies
visualized through hidden variables can be a reasonable solution for exploratory
data analysis.
3
If
n
is odd,all the hidden variables except one can have two child nodes.In this
case,the last one has only one child node.
Learning Hierarchical Bayesian Netw
orks for Large-Scale Data Analysis 673
3 Learning the Hierarchical Bayesian Network
Assume that we have a training dataset for
Y
consisting of
M
examples,i.e.,
D
Y
=
{
y
1
,
y
2
,...,
y
M
}
.We could describe our learning objective (log likelihood)
as follows.
L
(
Θ
S
h
,S
h
)=
M

m
=1
log
P
(
y
m
|
Θ
S
h
,S
h
)=
M

m
=1
log

Z
P
(
Z
,
y
m
|
Θ
S
h
,S
h
)
,
(2)
where

Z
means summation over all possible configurations of
Z
.General ap-
proach to finding maximum likelihood solution with missing variables,i.e.,the
expectation-maximization (EM) algorithm,is not applicable here because the
number of missing variables amounts to several thousands.The large number of
missing variables would render the solution space infeasible.
Here,we propose an efficient algorithm for maximizing the lower bound of
Eqn.(2).The lower bound for the likelihood function is derived by Jensen’s
inequality as follows.
M

m
=1
log

Z
P
(
Z
,
y
m
|
Θ
S
h
,S
h
)

M

m
=1

Z
log
P
(
Z
,
y
m
|
Θ
S
h
,S
h
)
.
(3)
Further,the term for each example,
y
m
,in the above equation can be decom-
posed as follows.

Z
log
P
(
Z
,
y
m
|
Θ
S
h
,S
h
)=

Z
log
P
(
y
m
|
Θ
S
h
Y
,S
h
)
·
P
(
Z
|
y
m
,
Θ
S
h
\
Θ
S
h
Y
,S
h
)
=
C
0
·
log
P
(
y
m
|
Θ
S
h
Y
,S
h
)+

Z
log
P
(
Z
|
y
m
,
Θ
S
h
\
Θ
S
h
Y
,S
h
)
,
(4)
where
C
0
is a constant which is not related to the choice of
Θ
S
h
and
S
h
.In
Eqn.(4),the parameter sets
Θ
S
h
Y
and
Θ
S
h
\
Θ
S
h
Y
can be learned separately
given
S
h
.
4
Our algorithm starts by learning
Θ
S
h
Y
and the substructure of
S
h
related to only the parents of
Y
.After that,we fill missing values for the variables
in the first hidden layer,making the training dataset for
Z
1
.
5
Now,Eqn.(4) can
be more decomposed as follows.
C
0
·
log
P
(
y
m
|
Θ
S
h
Y
,S
h
)+

Z
log
P
(
Z
|
y
m
,
Θ
S
h
\
Θ
S
h
Y
,S
h
)
=
C
0
·
log
P
(
y
m
|
Θ
S
h
Y
,S
h
)+
C
1
·
log
P
(
z
1
m
|
Θ
S
h
Z
1
,S
h
)
+

Z
\
Z
1
log
P
(
Z
\
Z
1
|
y
m
,
z
1
m
,
Θ
S
h
\{
Θ
S
h
Y
,
Θ
S
h
Z
1
}
,S
h
)
,
(5)
4
In this paper,the symbol ‘
\
’ denotes set difference.
5
Because hidden variables are all missing,this procedure is likely to produce hidden
constants by maximizing the likelihood function.We apply an encoding scheme for
preventing this problem,which will be described later.
674 K.-B.Hwang,B.-H.Kim,and B.-T.Zhang
Table 1.
Description of the two-phase learning algorithm for hierarchical Bayesian
networks.Here,hidden layer 0 means the observed layer and the variable set
Z
0
means
the observed variable set
Y
.
Input
D
Y
=
{
y
1
,
y
2
,...,
y
M
}
- the training dataset.
Output
A hierarchical Bayesian network
￿
S

h
,
Θ

S
h
￿
,maximizing
the lower bound of
￿
M
m
=1
log
P
(
y
m
|
Θ
S
h
,S
h
).
The First Phase
-For
l
=0to

log
2
n
−
1
- Estimate the mutual information between all possible
variable pairs,
I
(
Z
li
;
Z
lj
),in hidden layer
l
.
- Sort the variable pairs in decreasing order of mutual
information.
- Select

n
·
2

(
l
+1)

variable pairs from the sorted list
such that each variable is included in only one variable
pair.
- Set each variable in hidden layer (
l
+1) as the parent
of a selected variable pair.
- Learn the parameter set
Θ
S
h
Z
l
by maximizing
￿
M
m
=1
log
P
(
z
lm
|
Θ
S
h
Z
l
,S
h
).
- Generate the dataset
D
Z
l
+1
based on the current HBN.
The Second Phase
- Learn the Bayesian network structure inside hidden layer
l
by maximizing
￿
M
m
=1
log
P
(
z
lm
|
Θ
S
h
Z
l
,S
h
).
where
C
1
is a constant which is not related to the optimization.Then,we could
learn
Θ
S
h
Z
1
and related substructure of
S
h
by maximizing log
P
(
z
1
m
|
Θ
S
h
Z
1
,S
h
).
In this way,we could learn the hierarc
hical Bayesian network from bottom to
top.
We propose a two-phase learning algorithmfor hierarchical Bayesian networks
based on the above decomposition.In the first phase,a hierarchy for information
compression is learned.From the observed layer,we choose variable pairs shar-
ing common parents in the first hidden lay
er.Here,we should select the variable
pair with high mutual information for minimizing information loss.After deter-
mining the parent for each variable pair,missing values for the hidden parent
variable are filled as follows.First,we estimate the joint probability distribution
for the variable pair,
ˆ
P
(
Y
j
,Y
k
)(1

j,k

n
,
j

=
k
),from the given dataset
D
Y
.Then,we set the parent variable value for the most probable configuration
of
{
Y
j
,Y
k
}
as 0 and the second probable one as 1.
6
By this encoding scheme,a
parent variable could represent the two most probable configurations of its child
variables,minimizing the information loss.The parent variable value for other
two cases are considered as missing.Now,we could learn the parameters for
the observed variables,
Θ
S
h
Y
,using the standard EM algorithm.After learning
Θ
S
h
Y
,we could fill the missing values by probabilistic inference,making a com-
plete dataset for the variables in the first hidden layer.Now,the same procedure
6
Here,we assume that all variables are binary although our method could be extended
to more general cases.
Learning Hierarchical Bayesian Netw
orks for Large-Scale Data Analysis 675
can be applied for
Z
1
,learning the parameter set
Θ
S
h
Z
1
and generating a com-
plete dataset for
Z
2
.This process is iterated,building the hierarchical structure
and making the complete dataset for all variables,
{
Y
,
Z
}
.
After building the hierarchy,we lear
n the edges inside a layer when necessary
(the second phase).Any structural learnin
g algorithm for Bayesian network can
be employed because a comp
lete dataset is now given for the variables in each
layer.Table 1 summarizes the two-phase algorithm for learning HBNs.
4 Experimental Evaluation
4.1 Results on Synthetic Datasets
To simulate diverse situations,we expe
rimented with the datasets generated
fromvarious large-scale Bayesian networks having different structural properties.
They are categorized into scale-free [2
] and modular structures.All variables
were binary and local probability distributions were randomly generated.Here,
we show the results on two scale-free and modular Bayesian networks,consisting
of 5000 nodes.
7
They are shown in Fig.2(a) and 2(b),respectively.Training
datasets,having 1000 examples,were generated from them.
The first phase of HBN learning algorithm was applied to the training
datasets,building hierarchies.Then,the second phase was applied to the seventh
hidden layer,consisting of 40 hidden variables.The learned Bayesian network
structures inside the seventh hidden l
ayer are shown in Fig.2(c) and 2(d).
We examined the quality of information compression.Fig.3(a) shows the
mutual information value between parent nodes in the upper layer and child
nodes in the lower layer,averaged across a hidden layer.
8
Here,we can observe
that the amount of shared information between consecutive layers is more than
50%,
9
although it decreases as the level of hidden layer goes up.Interestingly,
the hierarchical Bayesian network can preserve more information in the case
of the dataset from modular Bayesian networks.To investigate this further,we
estimated the distribution of the mutual information (see Fig.3(b)).Here,we can
clearly observe that more in
formation is shared between parent and child nodes
in the case of the modular Bayesain network than in the case of the scale-free
Bayesian network.
Based on the above experimental resul
ts,we conclude that the hierarchical
Bayesian network can efficiently represent the complicated information contained
in a large number of variables.In addition,HBNs are more appropriate when the
true probability distribution assumes a modular structure.We conjecture that
this is because a module in the lower layer can be well represented by a hidden
node in the upper layer in our HBN framework.
7
Results on other Bayesian networks were similar although not shown here.
8
Here,mutual information values were scaled into [0
,
1],by being divided by the
minimum of the entropy of each variable.
9
It means that the scaled mutual information value is greater than 0
.
5.
676 K.-B.Hwang,B.-H.Kim,and B.-T.Zhang
Pajek
(a)
Pajek
(b)
H1
H2
H3
H4
H5
H6
H7
H8
H9
H10
H11
H12
H13
H14
H15
H16
H17
H18
H19
H20
H21
H22
H23
H24
H25
H26
H27
H28
H29
H30
H31
H32
H33
H34
H35
H36
H37
H38
H39
H40
Pajek
(c)
H1
H2
H3
H4
H5
H6
H7
H8
H9
H10
H11
H12
H13
H14
H15
H16
H17
H18
H19
H20
H21
H22
H23
H24
H25
H26
H27
H28
H29
H30
H31
H32
H33
H34
H35
H36
H37
H38
H39
H40
Pajek
(d)
Fig.2.
Scale-free (a) and modular (b) Bayesian networks,consisting of 5000 nodes,
which generated the training datasets.Th
e Bayesian network structures,inside the
seventh hidden layer,learned from the train
ing datasets generated from the scale-free
(c) and modular (d) Bayesian networks.These network structures were drawn by Pajek
software [3].
4.2 Results on a Real-Life Microarray Dataset
A real-life microarray dataset on the budding yeast cell-cycle [11] was analyzed
by hierarchical Bayesian networks.The dataset consists of 6178 genes and 69
samples.We binarized gene expression level based on the median expression of
each slide sample.Among the 6178 genes,we excluded genes with low informa-
tion entropy (
<
0
.
8).Finally,we analyzed a binary dataset consisting of 6120
variables and 69 samples.
General tendency with respect to the information compression was similar to
the case of synthetic datasets,although not shown here.In the seventh hidden
layer,consisting of 48 variables,we learned a Bayesian network structure (see
Fig.4).This Bayesian network compactl
y visualizes the original network struc-
ture consisting of 6120 genes.From the network structure,we can easily find a
set of hub nodes,e.g.,H2,H4,H8,H28,and H30.
Genes comprising these hub nodes and their biological role were analyzed.The
function of genes can be described by Gene Ontology (GO) [1] annotations.GO
Learning Hierarchical Bayesian Netw
orks for Large-Scale Data Analysis 677
0
1
2
3
4
5
6
7
8
0.45
0.5
0.55
0.6
0.65
0.7
0.75
0.8
Hidden layer
Average MI between parent and child nodes
Scale−free Bayesian network
Modular Bayesian network
(a)
0.5 0.6 0.7 0.8 0.9
02468101214
MI between parent and child nodes
Density
Scale−free Bayesian network
Modular Bayesian network
(b)
Fig.3.
Quality of information compression by hierarchical Bayesian network learning.
Mutual information between parent nodes in the upper layer and child nodes in the
lower layer (a) and its distribution (b).
H1
H2
H3
H4
H5
H6
H7
H8
H9
H10
H11
H12
H13
H14
H15
H16
H17
H18
H19
H20
H21
H22
H23
H24
H25
H26
H27
H28
H29
H30
H31
H32
H33
H34
H35
H36
H37
H38
H39
H40
H41
H42
H43
H44
H45
H46
H47
H48
Pajek
Fig.4.
The Bayesian network structure in the seventh hidden layer consisting of 48 vari-
ables.This network structure approximately represents the original network structure
consisting of 6120 yeast genes.Here,we can easily find some hub nodes,for example,
H2,H4,H8,H28,and H30.The network structure was drawn by Pajek software [3].
maps each gene or gene product to directly related GO terms,which have three
categories:biological process (BP),cellu
lar compartment (CC),and molecular
function (MF).We can conjecture the meaning of each hub using this annota-
tion,focusing on the BP terms related to the cell-cycle.For this task,we used
GO Term Finder (
http://db.yeastgenome.org/cgi-bin/GO/goTermFinder
),
which looks for significantly shared GO terms that are directly or indirectly
related to the given list of genes.The result are summarized in Table 2.The
closely located hub nodes,H4 and H8 (see Fig.4),share the function of cellular
678 K.-B.Hwang,B.-H.Kim,and B.-T.Zhang
Table 2.
Gene function annotation of hub nodes in the learned Bayesian network
consisting of 48 variables.The significance of a GO term was evaluated by examining
the proportion of the genes associated to this term,compared to the number of times
that term is associated with other genes in the genome (p-values were calculated by a
binomial distribution approximation).
Node name GO term Frequency p-value
H2 Organelle organization and biogenesis 18.7% 0.09185
H4 Cellular physiological process 74.0% 0.01579
H8 Cellular physiological process 74.2% 0.01357
H28 Response to stimulus 14.8% 0.00514
H30 Cell organization and biogenesis 29.3% 0.02804
physiological process.The genes in H3
0 share more specific function than H4
and H8,namely cell organization and biogenesis.The hub node H2 is related to
organelle organization and biogenesis,which is more specific than that of H30.
The genes in H28,the most crowded hub in the network structure,respond to
stress or stimulus such as nitrogen starvation.
5Conclusion
We proposed a new class of Bayesian networks for analyzing large-scale data
consisting of thousands of variables.The hierarchical Bayesian network is based
on hierarchical compression of information and constrained structural learning.
Through the experiments on datasets from synthetic Bayesian networks,we
demonstrated the effectiveness of hiera
rchical Bayesian network learning with
respect to information compression.Interestingly,the degree of information con-
servation was affected by structural pr
operty of the Bayesian networks which
generated the datasets.The hierarchical Bayesian network could preserve more
information in the case of modular networks than in the case of scale-free net-
works.One explanation for this phenomenon is that our HBN method is more
suitable for modular networks because a variable in the upper layer could well
represent a set of variables contained in a module in the lower layer.Our method
was also applied to analysis of real microa
rray data on yeast cell-cycle,consisting
of 6120 genes.We were able to obtain a reasonable approximation,consisting of
48 variables,on the global gene expression network.A hub node in the Bayesian
network consisted of genes with similar functions.Moreover,neighboring hub
nodes in the learned Bayesian network also shared similar functions,confirming
the effectiveness of our HBN method for real-life large-scale data analysis.
Acknowledgements
This work was supported by the Soongsil University Research Fund and by the
Korea Ministry of Science and Technology under the NRL program.
Learning Hierarchical Bayesian Netw
orks for Large-Scale Data Analysis 679
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