Correction
13 Aug 2013: Hu L, Wang L (2013) Correction: Using Consensus Bayesian Network to Model the Reactive Oxygen Species Regulatory Pathway. PLOS ONE 8(8): 10.1371/annotation/4bf3c893-ad02-4730-b775-fd83fd8d174c. https://doi.org/10.1371/annotation/4bf3c893-ad02-4730-b775-fd83fd8d174c View correction
Figures
Abstract
Bayesian network is one of the most successful graph models for representing the reactive oxygen species regulatory pathway. With the increasing number of microarray measurements, it is possible to construct the Bayesian network from microarray data directly. Although large numbers of Bayesian network learning algorithms have been developed, when applying them to learn Bayesian networks from microarray data, the accuracies are low due to that the databases they used to learn Bayesian networks contain too few microarray data. In this paper, we propose a consensus Bayesian network which is constructed by combining Bayesian networks from relevant literatures and Bayesian networks learned from microarray data. It would have a higher accuracy than the Bayesian networks learned from one database. In the experiment, we validated the Bayesian network combination algorithm on several classic machine learning databases and used the consensus Bayesian network to model the 's ROS pathway.
Citation: Hu L, Wang L (2013) Using Consensus Bayesian Network to Model the Reactive Oxygen Species Regulatory Pathway. PLoS ONE 8(2): e56832. https://doi.org/10.1371/journal.pone.0056832
Editor: Frank Emmert-Streib, Queen's University Belfast, United Kingdom
Received: May 19, 2012; Accepted: January 16, 2013; Published: February 15, 2013
Copyright: © 2013 Hu and Wang. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This research is supported by Special Science Foundation for the Doctoral Program of China (No. 200801831011) and Postdoctoral Science Foundation of China (No. 20100481053). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.
Competing interests: The authors have declared that no competing interests exist.
Introduction
Reactive Oxygen Species (ROS) are formed as by-products of normal metabolism of aerobic organisms, they can react with DNA and produce damage [1]. Cells protect themselves from ROS by detoxification mechanisms and repair mechanisms [2], [3]. Microarray is a powerful tool for measuring a large number of genes' expressions. Given the microarray expressions, it is possible to construct the regulatory pathway that organisms response to the oxidative stress directly.
An outstanding idea is the use of Bayesian network for representing regulatory pathway [4]–[7]. Bayesian network is a Directed Acyclic Graph (DAG) used for representing probabilistic relationships between variables. It was first proposed by Pearl [8], and Jensen [9] gave an intuitive definition. A lot of work has been done in the automatic learning of Bayesian network from database. Consequently, large numbers of Bayesian network learning algorithms based on different methodologies have been developed [10]–[13] and they have high accuracies in learning Bayesian networks from classic machine learning databases. However, when applying these algorithms to learn Bayesian networks from microarray data, the accuracies are low. Careful studies show that this is because the databases they used to learn Bayesian networks contain too few microarray data. On the other hand, microarray chip is expensive, it is difficult to obtain a large number of microarray data from one laboratory or one database, and a few hundred expression data can not guarantee a high learning accuracy.
To overcome this problem, we propose a consensus Bayesian network which is constructed by combining several Bayesian networks. This consensus Bayesian network is approximately equal to the Bayesian network learned from the database obtained by merging all these combined Bayesian networks' corresponding databases, then its equivalent database may have enough data and the accuracy can be improved. The main procedure of construction of consensus Bayesian network can be described as follow: (1) Review all relevant literatures and derive the Bayesian networks. (2) Search microarray expressions which are not used in relevant literatures and download them to learn Bayesian networks. (3) Combine all these Bayesian networks to construct the consensus Bayesian network.
Combination of Bayesian networks includes combination of graph models and aggregation of probability distributions [14]–[17]. Utz [18] proposed a method to combine many different Bayesian networks into an undirected graph, and each edge in the graph has a weight represents the frequency with which the edge occurs in the component networks. Zhang et al. [19] proposed a method for fusing Bayesian networks. They construct an initial network based on the union and intersection of the Bayesian networks, and then search for the structure which maximizes the scoring function(CH criterion). Our Bayesian network combination algorithm is based on the properties of probability. Due to probabilistic independence, Conditional Probability Tables (CPTs) can be extended, then corresponding nodes' CPTs can be changed into a same form and the aggregation function can be applied to these CPTs. After extending every corresponding CPTs, the combination of Bayesian networks changed into the aggregations of every corresponding nodes' CPTs if these Bayesian networks' variables' prior orders are consistent with each other. Some nodes' CPTs were extended previously, so they may have bogus parents after combination, then we should find them, delete the bogus edges and simplify the CPTs. The combination algorithm can also be applied to the combination of Bayesian networks defined over different variable sets by using the extension of Bayesian network.
was used in the experiment, a constructed ROS pathway was derived from the literature wrote by Hodges et al. [20] and 612 microarray expression data were downloaded from the Many Microbe Microarrays Database(M3D) [21]. 27 genes were identified from the EcoCyc [22] ROS detoxification pathway. A consensus Bayesian network using the 27 genes as variables was constructed by combining the Bayesian network from the literature and the Bayesian network learned from the 612 microarray expressions. For demonstrating the combination of Bayesian networks defined over different variable sets, we used a prediction program to find genes may be involved in the ROS pathway, learned a Bayesian network which using the 27 genes and the newly found genes as variables, and then combined this Bayesian network and the Bayesian network from the literature to construct a new consensus Bayesian network.
Results
Validation on classic machine learning databases
In order to validate whether the consensus Bayesian network constructed by combining Bayesian networks
and
is equivalent to the Bayesian network learned from the database obtained by merging the two Bayesian networks' corresponding databases
and
or not, 6 databases were downloaded from the UCI Machine Learning Repository (http://archive.ics.uci.edu/ml/), and the databases of ALARM net and Chest-clinic net were generated by the BN PowerConstructor. For each database
, we chose
samples (about
of the samples in
) randomly and used them as
, the rest samples in
were used as
. Two Bayesian networks
and
were learned from
and
, respectively. Consensus Bayesian network
was constructed by combining
and
. After that, another Bayesian network
used as a reference was learned from
,
was compared with
and the proportion of the number of identical edges between
and
to the total number of edges in
and
(similarity
) was computed. The program was run 100 times to compute the average similarity. All results of the experiments are shown in Table 1. Table 1 shows that all the average similarities are greater than
. So, consensus Bayesian network
is approximately equal to Bayesian network
. Although the combination algorithm is validated with 8 different databases and the types of data in these databases are very different, it doesn't affect the results. The Bayesian network learning algorithm just compute the distributions by counting the number of samples, and determine the relationships between the variables by analyzing the distributions. The Bayesian network combination algorithm is used to combine Bayesian networks and it doesn't involve the data. So, the type of data doesn't affect the validation.
Consensus Bayesian network is approximately equal to Bayesian network
, then we can view
's database
as
's equivalent database, and
have more samples than
or
. So, the use of consensus Bayesian network helps to solve the problem of lack of data in partial databases and the accuracy can be improved. The true structures of ALARM net and Chest-clinic net are known. Then we compared the learned networks with the known networks, the results are shown in Table 2. Table 2 shows that
has a higher accuracy than
or
.
Construction of consensus Bayesian network for modeling 
's ROS pathway
The consensus Bayesian network is constructed by combining Bayesian networks derived from literatures and Bayesian networks learned from microarray data. First, relevant literatures were reviewed and a ROS pathway was derived from the literature wrote by Hodges et al.[20], denoted as . In the literature, 27 genes identified from the EcoCyc ROS detoxification pathway were chosen as variables and 305 microarray expressions were used to learn the Bayesian network. Second, microarray data was searched and a microarray expression build with 612 microarray expressions was downloaded from the M3D database. Then Bayesian network
which also uses the 27 genes as variables was learned from these microarray expressions. Finally, consensus Bayesian network
was constructed by combining these two Bayesian networks, and the result is shown in Figure 1. In the combination program, we take weights
,
and threshold
.
27 genes were identified from the EcoCyc ROS detoxification pathway.
A novel prediction algorithm based on the computation of mutual information was developed to identify genes which are strongly associated with a particular gene in the regulatory pathway. If is a gene in the regulatory pathway, gene
is strongly associated with
, then
may work together with
and also be involved in the pathway. The main procedure of this algorithm can be described as follow: assume set
includes all the known genes in the regulatory pathway, and set
includes the rest genes of the organism. Choose one gene
in
, for each gene
, compute the mutual information
, if
, it means gene
is related to gene
, then
may be involved in the pathway too. The program is ended until every gene in
has been tested.
27 genes identified from the EcoCyc ROS detoxification pathway were used as set , while the rest genes in
were used as set
. The program found 4 genes may be involved in the ROS pathway, and the results are shown in Table 3. A new Bayesian network
using the 31 (27+4) genes as variables was learned from the 612 microarray expressions.
contains more genes than
, so
was extended into
. Then a new consensus Bayesian network
was constructed by combining
and
, and the result is shown in Figure 2.
27 genes were identified from the EcoCyc ROS detoxification pathway, while genes ,
,
,
were identified by the prediction program.
Discussion
In the discussion, we address this question: does the Bayesian network learned from microarray expressions match with a known regulatory pathway?
Before answering this question, we carried out an experiment. The procedure of the experiment can be described as follow: assume that includes all of the genes of
, and then we construct an undirected graph
, where
. Let
be the largest connected subgraph of
. Then
of the genes in
were included in
. Mutual information
means genes
and
are interacted, so this phenomenon shows that almost all genes in
are related directly or indirectly. We can infer that some genes may be involved in different regulatory pathways, simultaneously. Otherwise, if there is no gene be involved in more than one regulatory pathway, that is, the regulatory pathways in
have no intersection, then we can't observe the phenomenon that thousands of genes related directly or indirectly. On the other hand, before microarray measurements, the
was alive, so almost all of the regulatory pathways of
were at work. Then although two genes must be interacted if there is a directed edge between them in the Bayesian network, it is hard to determine the directed edge belongs to which regulatory pathway. For example, there is a directed edge between
and
in
(Figure 1), then there must be an interaction between
and
. They are involved in the regulation of transcription (EcoCyc database) and this biological process was working when measuring the expressions of these genes using microarray, therefore, the existence of
maybe due to that they are regulating the transcription. However, the ROS detoxification pathway (EcoCyc database) also contains
and
, then the existence of
maybe due to that they are regulating the response to the oxidative stress. So, it is hard to determine the directed edge
belongs to which regulatory pathway. If there is no edge between two genes in the Bayesian network, then the two genes are not interacted directly in any regulatory pathway. So, if a known regulatory pathway contains
genes and we use these
genes as variables to learn a Bayesian network from microarray expressions. Then all of the interactions between the
genes are contained in the Bayesian network, however, some of these interactions may not contained in this regulatory pathway. This means the regulatory pathway is a subgraph of the Bayesian network. Although the Bayesian network is not equivalent to the regulatory pathway, it still has important significance. With its guidance, the number of biological experiments could be greatly reduced when modeling a regulatory pathway.
Methods
Data preprocessing
The algorithms can only process discrete data in this paper. However, the 612 microarray expression data of downloaded from the M3D database are continuous. Then expression data for each gene was discretized using a maximum entropy approach which uses three equally-sized bins (q3 quantization). And the genes' expressions were divided into three categories: underexpressed, normal, overexpressed.
Usually, Bayesian networks derived from literatures only have a structure, then we have three ways to obtain the parameters: (1) If the program of the learning algorithm is available on the internet, then both the structure and the parameters of the Bayesian network can be obtained by run the program directly. (2) If the microarray data used in the literatures were collected in a database available on the internet, then we can download these microarray data to learn the parameters. (3) Sometimes the corresponding database is unable to be found, or the Bayesian network is not learned form database, but constructed by biological experiments directly. Then distribution for each node can be estimated by analyzing the genes' special characteristics and the relationships between genes.
Bayesian network
A Bayesian network defined over a variable set can be represented as a pair
, where
is a DAG and each directed edge in the DAG represents a dependence,
is a group of CPTs and each node in the DAG has a CPT. Usually,
is called Bayesian network's structure and can be represented as a pair
, where
is the edge set;
is called Bayesian network's parameter.
is a directed acyclic graph, that is, the nodes in
have a topological order, and we call it prior order. Let
and
be two Bayesian networks and their DAGs are
and
, respectively, then
and
's variables' prior orders are consistent with each other if
is acyclic. Let
be a node in
,
's direct precursor nodes are called its parents, denoted as
, then
's CPT represents the conditional probability
. Suppose we have the CPT of
as shown in Figure 3(c), it shows
is a parent of
and
means
. Assume that CPT
represents
,
represents
and
represents
. Then
and
are two tables with the same structure and the conditional probabilities in the corresponding positions of the two tables represents the same conditional probability, so we say they have a same form. While
and
do not have a same form.
Bayesian network learning algorithm
Usually, Bayesian network is learned from database, it represents the probabilistic relationships between the variables in the database. So, a Bayesian network matches with a database, and we call this database Bayesian network's corresponding database. Bayesian network learning includes structure learning and parameter learning. We use an information-theory based learning algorithm proposed by Cheng et al. [11] to learn Bayesian network's structure in this paper.
Dependence between two variables can be quantitatively computed by using mutual information. Mutual information between two variables
and
can be defined as:
(1)
where ,
are the expression values of
and
, respectively. Mutual information is non-negative, it means
.
holds if and only if
and
are independent. Given a threshold
(
),
and
are related if
. Similarly, conditional mutual information
can be defined as:
(2)
Then the main procedure of Cheng's Bayesian network structure learning algorithm can be described as follow:
Step 1. Create initial undirected graph. A Maximum Weight Span Tree (MWST) [23] is used as the initial graph. Let be an undirected edge list, where
is the variable set. Sort
in descending order of mutual information. For each
, add it into the undirected graph(and delete it from
) if it doesn't form a circle. End this loop until the graph contains
edges.
Step 2. Add edges. Assume that set contains all the nodes which are in the paths between
and
and in the neighborhood of
, simultaneously.
represents one of sets
and
which contains less nodes. For each
, add it into the undirected graph(and delete it from
) if
holds.
Step 3. Remove redundant edges. For each edge in the undirected graph, delete it if
holds.
Step 4. Determine edges' directions. For each , direct them
if
(3)
holds, where threshold . Some undirected edges' directions can be determined by using Bayesian network's acyclic property. For the rest undirected edges, use the local Minimal Description Length (MDL) score [24] to choose the direction which makes the MDL score more smaller.
In Bayesian network parameter learning, the following equation is used to compute the conditional probabilities in each node's CPT:(4)
where is the number of samples satisfies
in the database.
Extension and simplification of CPT
Corollary 1.
Given ,
and any other variable
, then
(
) holds if
and
are independent given
.
Suppose we have the CPT of node as shown in Figure 3(a), it can be extended into the form as shown in Figure 3(b) if
and
are independent of each other. Since
and
are independent,
can not affect the distribution of
, then for
,
holds. According to that, two CPTs of a same node in different Bayesian networks can be extended into a same form, and then can be aggregated even if the node does not have a same parent set in these Bayesian networks. Specifically, for a node
, and its parent sets are
and
in
and
, respectively. Then the two CPTs of
do not have a same form, and the aggregation function can't be applied (See the CPTs of
shown in Figure 3(b) and Figure 3(c), they have a same form, then the aggregation function can be applied to aggregate the conditional probabilities in the corresponding position of the two CPTs, and the aggregation function can't be applied to aggregate the CPTs shown in Figure 3(a) and Figure 3(c)). However, we can take
as the parent set and extend both the CPTs of
in
and
into form
, then the two CPTs of
have a same form and the aggregation function can be applied. This means we also view the nodes in
as the parents of
in
, although they are not real parents and do not affect
's conditional probability. We call these parents bogus parents and the directed edges between a node and its bogus parents bogus edges. As shown in Figure 4(c),
is a bogus parent of
, and
is a bogus edge.
Assume that weights ,
, and we have two Bayesian networks as shown in
and
. The CPTs of
in the two Bayesian networks do not have a same form, so they need to be extended. After extending the CPTs, the two Bayesian networks' structures and every corresponding CPTs' forms are completely the same(as shown in
and
, the dashed edges represent bogus edges), and then the aggregation function can be applied to aggregate the conditional probabilities in corresponding positions of each corresponding CPTs. For example,
. In the combined Bayesian network as shown in
, we need to use variance to test
's two parent nodes,
,
, so
is a bogus parent. Then the CPT of
need to be simplified and the bogus edge
should be deleted. The consensus Bayesian network is shown in
.
Theorem 2.
Given variables and
,
is independent of
if the conditional probability of
does not change when
takes different values.
Proof.
Assume that the number of expression values of is
and for
,
. Then for
, we have:
So, , then
and
are independent.
End of the proof.
Corollary 2.
Given ,
and any other variable
,
is independent of
given
if the conditional probability of
does not change when
takes different values (only
changes).
Theorem 2 and Corollary 2 can be used to determine whether two nodes are independent of each other or not. Suppose we have the CPT of node as shown in Figure 3(c), if
,
or they are approximately equal, it deduces that
and
are independent, and
is not the parent node of
. Then the CPT of node
can be simplified into the form as shown in Figure 3(d). Conditional probabilities in the CPT of
are discrete values, then variance can be used to determine whether the conditional probability of
changes or not when
takes different values. Assume that
's parent set is
. First, compute each variance
of the conditional probabilities satisfy
,
and
takes different values. Second, compute the average variance
of all
when
and
take different values. Given a threshold
(
), if
, it means the conditional probability of
almost does not change when
takes different values, then
and
are independent and
is not the parent node of
. In the combination algorithm, CPTs were extended previously, then some nodes may have bogus parents after the aggregation of CPTs. However, we can use this method to find them, and then simplify the CPTs and delete the bogus edges. Threshold
can be selected by using domain knowledge. Specifically, we have
if variables
and
are related, and
if variables
and
are independent. And then we have
if there is
pair of variables related, and
if there is
pair of variables independent each other. So we have
.
Aggregation function
Assume that the conditional probability of node in Bayesian networks
and
are
and
, respectively. Then in the consensus Bayesian network, the conditional probability of
can be computed using the following equation:
(5)
where is the weight of
and
is the weight of
. Weight
is a positive integer representing a belief to the Bayesian network.
means
is more reliable than
;
means
is absolutely reliable.
Next, we would like to discuss why we choose this aggregation function. The combination of Bayesian networks must satisfies this property: the consensus Bayesian network constructed by combining Bayesian networks
and
is equivalent to the Bayesian network learned from the database obtained by merging the two Bayesian networks' corresponding databases
and
. Then the aggregation function should satisfy it too. Assume that node
is not the parent of
in any Bayesian network, then in the consensus Bayesian network,
can not be the parent of
. Then CPTs of
in different Bayesian networks after extension not only have a same form, but also contains all of
's possible parent nodes. So, we needn't consider the nodes which are not included in the parent set of
when aggregating the CPTs. When computing the conditional probability of one node in Bayesian network, Equation (4) is used. The conditional probability of
in Bayesian networks
and
are
and
, respectively. Assume that the number of samples satisfy
in
is
, then the number of samples satisfy
and
in
is
; the number of samples satisfy
in
is
, then the number of samples satisfy
and
in
is
. So, the conditional probability of
in the Bayesian network learned from the database obtained by merging
and
is:
(6)
On the other hand, samples satisfy in
and in
obey the same distribution. So, we have:
(7)
where and
are the total numbers of samples in
and
, respectively. Then the conditional probability of
changed to be:
(8)
Total numbers of samples in databases are unable to be known sometimes, so we use the weights of the Bayesian networks instead of them, then Equation (8) changed into Equation (5). In the experiment, we still use the total numbers of samples as they are already known.
Combination of Bayesian networks
If two Bayesian networks are defined over the same variable set and their variables' prior orders are consistent with each other, then they can be combined using the method described as follow:
Step 1. Extend every corresponding CPTs in the two Bayesian networks into same form. Then the structures of the two Bayesian networks are completely the same(although some of their edges are bogus edges).
Step 2. Use the aggregation function to aggregate the conditional probabilities in the corresponding positions of each corresponding CPTs.
Step 3. In each CPT after aggregation, compute variance for each parent node
, determine whether
holds or not to judge node
is a bogus parent or not, then simplify the CPT and delete the bogus edge if
holds.
After simplifying the CPTs and deleting the bogus edges, the consensus Bayesian network is obtained. Figure 4 shows an example of combination of two Bayesian networks. However, Bayesian networks' variables' prior orders do not always consistent with each other, then it needs to reverse some directed edges sometimes. The principle of reversal is to ensure that the Bayesian network after reversal is equivalent to the original Bayesian network.
Extension of Bayesian network
Sometimes the Bayesian networks going to be combined may not defined over the same variable set, then they need to be extended. Specifically, given two Bayesian networks and
with their variable sets satisfy
and
, if their variables' prior orders are consistent with each other,
can be extended into
using the method described as follow:
Step 1. Extend 's DAG
into
. Let
, and then add all the directed edges satisfy
into graph . These added edges are not in
originally, so we call them extended edges.
Step 2. Compute each node's CPT. For a node , if
, then its CPT is the same as the CPT of
in
; if
and there is no directed edge satisfies
, then its CPT is the same as the CPT of
in
; if
and has directed edges satisfy
, in this case, there are three possible situations may appeared in the extended Bayesian network as shown in Figure 5. Then the conditional probabilities of
in these three situations can be computed using the following equations, respectively:
Where ,
,
and
may be two nodes or two node sets with each node in them has a directed edge point to
. In
, solid lines represent the edges in
originally, and dashed lines represent the extended edges. The undirected edge
represents one of these three cases: (1) directed edge
; (2) directed edge
; (3)
and
is disconnect.
In Figure 5(a)(9)
where is the number of expression values of
.
and
can be computed using the standard Bayesian network inference algorithm [25] in
, while
is already known in
.
In Figure 5(b)(10)
In Figure 5(c)(11)
If and
are disconnect in
, it can only deduce that
and
are independent given
, however, it doesn't affect the conditional probabilities
and
, then the conditional probability of
can be computed using Equation (9). If both
and
,
and
are disconnect in
, it deduces
and
are independent, then the conditional probability of
can be computed using Equation (10). After obtaining every node's CPT in
, the extension of Bayesian network
is finished.
After extending into
and
into
,
and
are defined over the same variable set
, and then they can be combined using the combination algorithm.
Author Contributions
Conceived and designed the experiments: LDH LMW. Performed the experiments: LDH LMW. Analyzed the data: LDH LMW. Wrote the paper: LDH.
References
- 1. Ramotar D, Popoff SC, Gralla EB, Demple B (1991) Cellular role of yeast apn1 apurinic endonuclease/3′-diesterase: repair of oxidative and alkylation dna damage and control of spontaneous mutation. Molecular and Cellular Biology 11: 4537–4544.
- 2. Demple B, Harrison L (1994) Repair of oxidative damage to dna: enzymology and biology. Annual Review of Biochemistry 63: 915–948.
- 3. Bohr VA, Dianov GL (1999) Oxidative dna damage processing in nuclear and mitochondrial dna. Biochimie 81: 155–160.
- 4. Friedman N, Linial M, Nachman I, Pe'er D (2000) Using bayesian networks to analyze expression data. Journal of Computational Biology 7: 601–620.
- 5. Irene M, Jeremy D, David P (2002) Modelling reguratory pathways in e. coli from time series expression profiles. Bioinformatics 18: S241–S248.
- 6. Beal M, Falciani F, Ghahramani Z, Rangel C, Wild D (2005) A bayesian approach to reconstructing genetic regulatory networks with hidden factors. Bioinformatics 21: 349–356.
- 7. Bansal M, Belcastro V, Ambesi-Impiombato A, di Bernardo D (2007) How to infer gene networks from expression profiles. Molecular Systems Biology 3: 78.
- 8. Pearl J (1986) Fusion, propagation and structuring in belief networks. Artificial Intelligence 29: 241–288.
- 9.
Jensen F (2001) Bayesian networks and decision graphs. New York: Springer.
- 10. Cooper GF, Herskovits E (1992) A bayesian method for the induction of probabilistic networks from data. Machine Learning 9: 309–347.
- 11. Cheng J, Greiner R, Kelly J, Bell D, Liu WR (2002) Learning bayesian networks from data: an information-theory based approach. Artificial Intelligence 137: 43–90.
- 12. Tsamardinos I, Brown LE, Aliferis CF (2006) The max-min hill-climbing bayesian network structure learning algorithm. Machine Learning 65: 31–78.
- 13. Cano A, Masegosa A, Moral S (2011) A method for integrating expert knowledge when learning bayesian networks from data. IEEE transactions on systems, man, and cybernetics 41: 1382–1394.
- 14. Wong SM, Butz CJ (2001) Constructing the dependency structure of a multiagent probabilistic network. IEEE Transactions on Knowledge and Data Engineering 13: 395–415.
- 15. Yang ZQ, Wright RN (2006) Privacy-preserving computation of bayesian networks on vertically partitioned data. IEEE Transactions on Knowledge and Data Engineering 18: 1253–1264.
- 16. Pavlin G, de Oude P, Maris M, Nunnink J, Hood T (2010) A multi-agent systems approach to distributed bayesian information fusion. Infromation Fusion 11: 267–282.
- 17. Sagrado J, Moral S (2003) Qualitative combination of bayesian networks. International Journal of Intelligent Systems 18: 237–249.
- 18. Utz CM (2010) Learning ensembles of bayesian network structures using random forest techniques. Master Thesis of University of Oklahoma
- 19. Zhang Y, Yue K, Yue M, Liu W (2011) An approach for fusing bayesian networks. Journal of Information and Computational Science 8: 194–201.
- 20. Hodges AP, Dai DJ, Xiang ZS, Woolf P, Xi CW, et al. (2010) Bayesian network expansion identifies new ros and biofilm regulators. PLoS ONE 5: e9513.
- 21. Faith JJ, Driscoll ME, Fusaro VA, Cosgrove EJ, Hayete B, et al. (2008) Many microbe microarrays database: uniformly normalized affymetrix compendia with structured experimental metadata. Nucleic Acids Research 36: D866–D870.
- 22. Keseler IM, Collado VJ, Gama CS, Ingraham J, Paley S (2005) Ecocyc: a comprehensive database resource for escherichia coli. Nucleic Acids Research 33: D334–D337.
- 23. Chow CK, Liu CN (1968) Approximating discrete probability distributions with dependence trees. IEEE Transactions on Information Theory 14: 462–467.
- 24. Lam W, Bacchus F (1994) Learning bayesian belief networks: an approach based on the mdl principle. Computational Intelligence 10: 269–293.
- 25.
Zhang LW, Guo HP (2006) Introduction to Bayesian network. Peking: Science Press.