Cluster-Based WSA for Decision Support Bayesian Systems: Case of
Prognostic in Maintenance Management
Imen Ben Brahim
1,2
, Sid-Ali Addouche
1
, Abderrahman El Mhamedi
1
and Younes Boujelbene
2
1
QUARTZ Laboratory EA 7393, IUT of Montreuil, University of Paris 8,
140, rue de la Nouvelle-France, 93100 Montreuil, France
2
URECA, University of Sfax, Airport Road Km4, 3018 Sfax, Tunisia
Keywords:
Elicitation, Bayesian Network, Weighted Sum Algorithm, Clustering, Decision Support.
Abstract:
A knowledge representation and reasoning from data have produced many models. Probabilistic graphical mo-
dels, specifically the Bayesian Network (BN) have proved its worth. It is considered to be a very useful tool for
representing uncertain knowledge and decision-making support. This presupposes availability of knowledge
problem in the conditional probabilities form. However, one is often in a critical situation because of data
are insufficient, partially unavailable or heterogeneous. Developing methods and techniques to reconstruct the
corpus of data needed for decision making, especially via BN is called the ”knowledge elicitation”. Several
elicitation methods exist but they are not always applicable, too demanding in expert knowledge or presenting
limits. The most generic and useful is the Weighted Sum Algorithm (WSA) but it presents two major issues
concerning the compatible parental configuration. In the present paper, we discuss what the literature propo-
ses for the first one, then we develop the solution for the second and validate it via a case of pump failure
prognostic tool based on Bayesian support decision.
1 INTRODUCTION
A knowledge representation and reasoning from data
have produced many models. Probabilistic graphi-
cal models, specifically the BN, initiated by Pearl
in the 1980s, have proved its worth. It is conside-
red to be a very useful tool for representing uncertain
knowledge and reasoning from incomplete informa-
tion. It also allows providing effective solutions with
compact graphical representations of real problems in
short time. In general, using as a decision support tool
presupposes availability of problem knowledge in the
conditional probabilities form. However, in many real
applications, we are often in a critical situation where
data are insufficient, partially unavailable or heteroge-
neous. Indeed, to definitively establish a conditional
probability table (CPT), availability of a large quan-
tity of data is required. In addition, the descriptive
knowledge of a problem is sometimes qualitative and
we have to transform them to quantitative knowledge
in order to build the related CPT. In these situations,
learning parameters of BN using expert knowledge to
estimate the conditional probabilities is the most reli-
able way. Developing methods and techniques to re-
construct the corpus of information needed for deci-
sion making, especially via BN is called the ”know-
ledge elicitation”. This is a branch of artificial intelli-
gence and most of the elicitation methods come from
Knowledge management techniques and they are de-
tached from the tool used to guide the decision ma-
ker. Among these methods is the Weighted Sum Al-
gorithm (WSA) which is for all purpose. This method
allows us to reduce the information number to elicit
with the expert but in the field of application, it pre-
sents gaps concerning the compatible parental confi-
gurations. One of them has no scientific contribution
thus it has been solved in this paper by proposing a
clustering approach.
This paper is organized as follows: in the next
section, we will introduce the BN, its structure and
parameters. In section 3, we will classify and discuss
various Bayesian-based elicitation methods found in
literature. In section 4, we will deal with the WSA
and its limits. Then, we will propose our clustering
approach as a method to remedy the scientific issue
encountered. In section 5, we will apply our method
on the ”Shutdown Pump” case study and we are going
to use some tools to facilitate the task for the expert.
Then, we will compare our method with WSA and the
Raw method of elicitation and give thereafter our re-
sults. Finally, in section 6, we will summarize the pa-
per and draw some conclusions about further works.
Ben Brahim I., Addouche S., El Mhamedi A. and Boujelbene Y.
Cluster-Based WSA for Decision Support Bayesian Systems: Case of Prognostic in Maintenance Management.
DOI: 10.5220/0006878502220230
In Proceedings of the 15th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2018), pages 222-230
ISBN: 978-989-758-321-6
Copyright
c
2018 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved
2 BAYESIAN NETWORK
A BN according to (Pearl, 2014) is a graphical model
representing the joint probability distribution P(X) on
a set of random variables X = {X
1
, ..., X
n
} defining
probabilities P ∈ [0,1] for each possible state (x
1
, , x
n
)
∈ X
1,dom
, ..., X
n,dom
where X
i,dom
is the domain of de-
finition of each variable X
i
. It is a directed acyclic
graph, in which nodes represent random variables,
arcs symbolize the relationships between these varia-
bles and by the set of CPTs of each node in the graph
given its parents. They encode the joint probability
over all the nodes as the product of these conditional
probabilities. The joint probability distribution on all
the variables X of this model is written as follows:
P(X
1
, ..., X
n
) =
n
∏
i=1
(P(X
i
|pa(X
i
))) (1)
2.1 Structure and Parameters
BN is a mix of probability and graph theories. In ot-
her words, it require a combination between expert
knowledge and data (Fenton and Neil, 2012). But, it
is not very common to combine both sources of in-
formation. In practice, many BNs models have been
learnt only with data however others have been built
solely on expert knowledge. It can be explained by
the difficulty to combine knowledge with data. BN
users should have a strong background in both data
mining and expert systems, as well as to have access
to, and time for, the actual domain expert elicitation
(Constantinou et al., 2016). For building a BN, we
should:
• Determining the structure of the network
According to (Na
¨
ım et al., 1999), the human interven-
tion in the first step of construction is important. We
have to determine all the variables that characterize
the system. Then we must specify the states of each
variable, that is to say all of its possible values. After
identifying the different variables and their state spa-
ces, we proceed thereafter to identify links between
these variables. In short, the variables involved (or
parent) and effect variables (or child) must be defined
(Figure 1).
Figure 1: Example of Bayesian network.
• Determining the conditional probabilities tables
Once the BN is established, we assign to each state of
each node (Variable) a conditional probability given
the statements of the parents to determine CPTs as-
sociated with different nodes. According to (Nguyen,
2007), expert knowledge on variable probability dis-
tributions is included in the model. Specifically, there
are two cases according to the position of a variable
X
i
in the BN: X
i
has no parent variable: experts must
identify the marginal distribution X
i
(See Table 1).
Table 1: CPT of X
1
.
X
1
x
1
... x
n
P(X
1
= x
1
) ... P(X
1
= x
n
)
X
i
has parental variables: experts must express
the dependence of X
i
function of the parent variable,
either by means of conditional probabilities, or by de-
terministic equation, which subsequently allows the
calculation of conditional probabilities (See Table 2).
Table 2: CPT of X
n
.
X
n
X
1
x
1
... x
n
x
1
P(X
n
= x
1
|X
1
= x
1
) ... P(X
n
= x
1
|X
1
= x
n
)
... ... ... ...
x
n
P(X
n
= x
n
|X
1
= x
1
) ... P(X
n
= x
n
|X
1
= x
n
)
2.2 Parameter Sourcing Problem
A BN’s direct acyclic graph and CPTs can be learnt
automatically from data, or by domain expert elici-
tation, or using the combination of both. However, in
most cases we do not found the necessary information
and data is not available for the determination of pro-
babilities. So, we need expert knowledge as source of
probabilities. Using expert’s beliefs to assign conditi-
onal probabilities is called expert elicitation.
According to (Cooke and Goossens, 2000), the
main steps of the elicitation process are:
• determine what the experts need to elicit,
• select experts,
• expert elicitation, where the experts may use an
elicitation method to assign probabilities,
• if there are several experts, combine their asses-
sments,
• document the process and the result.
We suppose that we have a simple BN, referred to
as a naive BN, where there is just one child and two
parent nodes with four and three states respectively.
The number of probabilities required for each node is
represented by equation 2.
NP = (m − 1)
k
∏
i=1
n
i
(2)
Where NP = the number of probabilities needed to
build a CPT of the child node ; m = the number of
states of the child node; and n = number of states of
parent node i = 1...k.
If a node has no parents, the product term drops
from the equation (it does not take the value of zero)
and only the prior probabilities (m − 1) are needed.
In its simplest form with all nodes binary, NP = 2
k
(Tang and McCabe, 2007). So and in this case, the
expert must elicit to us (4−1) ×3
2
= 27 probabilities
to fill the CPT of the child node.
Due to the structure of a BN the number of proba-
bilities in CPT grows exponentially with the number
of parent nodes and their states related to that CPT.
For example, if we add another parent node with 4
states and an additional state to other nodes (parent
and child). Then, CPT size would increase from 27
to 256 parameters. Thus, the elicitation becomes ex-
tremely laborious and very time consuming. For an
expert, it can be particularly difficult to assign proba-
bilities for events that are very rare. Therefore, diffe-
rent methods to help the expert and to systematize the
elicitation have been developed (Knochenhauer et al.,
2013) which will be detailed in the next section.
3 KNOWLEDGE ELICITATION
FOR DECISION SUPPORT
When developing BNs for practical applications, we
must incorporate expert knowledge of factors which
are important for decision analysis where historical
data is unavailable or difficult to obtain (Constantinou
et al., 2016). The involvement of expert elicitation
is a big challenge. First, the design of the elicitation
process itself, which includes the determination of ex-
perts and elicitation techniques, might be a daunting
process (Kuhnert et al., 2010). Moreover, the deman-
ding of large number of conditional probability en-
tries not only put great workload on the experts but
also poses challenges to the quality or consistency of
the elicited result. Therefore, as to experts elicitation,
many of the previous literature were working on two
aspects: one is to reduce the burden of the experts
by reducing the number of conditional probabilities
to elicit while the other is to facilitate the elicitation
of individual probability entry (Knochenhauer et al.,
2013). The difficulties are related to expert reliabi-
lity and pertinence. By the way and to facilitate the
elicitation, it is possible to provide expert tools lin-
king qualitative and quantitative concepts to associate
a probability to the various events.
3.1 Facilitation Methods
We found various methods to facilitate the elicitation
when we should elicitate one probability like Gamble
Like Methods originate from the standard gamble in-
troduced by (Von Neumann and Morgenstern, 2007)
as an indirect method for utility elicitation. The basic
idea behind a gamble like method is that the expert is
presented with a choice between two lotteries (Burg-
man et al., 2006). In addition, there is Probability
Wheel method which is an indirect method that is not
influenced by risk attitudes. It is usually a circle di-
vided into two sections. The sections are altered until
the expert believes he can spin a pointer and the pro-
bability that it will stop in a section is equivalent to the
probability being assessed (Renooij, 2001). Also, we
found Probability Scale which is the best known and
one of the various easiest methods to work. The basic
idea of this scale is to facilitate to the expert the deter-
mination of probabilities in CPTs by allowing them to
use information both textual and numeric to assign a
level of achievement to a particular assertion (Druzdel
and Van Der Gaag, 2000) and (Renooij and Witteman,
1999).
3.2 Reduction Methods
For reducing the number of conditional probabilities
to elicit, we present five of the most frequently used
methods:
1. Node divorcing which allows as to simplify the
model structure by introducing a mediating node
to reduce the number of parent nodes (Zhang and
Thai, 2016).
2. Likelihood method where the experts are asked
about the influence weight instead of the direct
conditional probability numbers (Kemp-Benedict,
2008).
3. EBBN method (an Elicitation method for Baye-
sian Belief Network) uses on piecewise linear in-
terpolation based on the ranks of the parent nodes’
states to determine the CPT (Wisse et al., 2008),
(Knochenhauer et al., 2013), (Mkrtchyan et al.,
2016).
4. Exploit the causal independence between the pa-
rent nodes. The most widely used way is to ex-
ploit the Noisy-OR role (for binary variables) and
its extension Noisy-Max (for nominal variables)
(Bolt and van der Gaag, 2010), (Kraaijeveld et al.,
2005). However, this method often make funda-
mental assumptions about the BN: assume that pa-
rent nodes must be independent, their states must
be ordered according to a criterion or express a de-
gree or gradient of a variable and that each node
must have an absent state. In reality, these as-
sumptions cannot often be satisfied.
5. Weighted Sum Algorithm(WSA) is a method
which solve many constraints of the previous met-
hod. In fact, what makes this algorithm special
compared to other techniques, is that it allows
us to ask experts questions that are easy to vi-
sualize and simulate (Baker and Mendes, 2010).
Also, using the WSA will make the number of as-
sessments of a CPT linear instead of exponential
(Das, 2004). That’s why our choice is converged
towards this method to more explore and apply it
to a case study. We develop it more in the next
section.
4 CLUSTER-BASED WSA
ELICITATION
The use of the WSA method is indeed practical in
terms of reducing the information to elicit but someti-
mes in some situations experts can’t objectively pro-
vide probabilistic data. In what follows, we will detail
the method as it has been presented with (Knochen-
hauer et al., 2013) to better explain and we will give
his limits. In next subsections, we explain WSA met-
hod and then solve the main of these limits for which
we don’t find literature contribution.
4.1 WSA Elicitation Method
According to notation used with (Knochenhauer et al.,
2013), this method consists of an algorithm that es-
timates the k
1
× ... × k
n
conditional probabilities :
P(X
c
= x
m
c
|X
p
1
= x
j
1
p
1
, X
p
2
= x
j
2
p
2
, ..., X
p
n
= x
j
n
p
n
) that
form a CPT. With X
c
as the child node with l states
and {X
p
i
}
n
i=1
as the parent nodes with k
i
states each
the algorithm takes the following form:
P(x
m
c
|x
j
1
p
1
, ..., x
j
n
p
n
) =
n
∑
i=1
w
i
P(x
m
c
|{Comp(X
p
i
= x
j
i
p
i
)})
(3)
where m = 1, 2, ..., l and j
i
= 1, 2, ..., k
i
.
This method requires the expert to elicit two
things:
• the relative weights w
1
, ..., w
n
for the parent no-
des, where 0 ≤ w
i
≤ 1 and
∑
n
i=1
w
i
= 1.
• the k
1
+... +k
n
probability distributions over X of
P(x
m
c
{Comp(X
p
i
= x
j
i
p
i
)}) (4)
for compatible parental configurations.
Compatible parental configurations refer to the term
{Comp(X
p
i
= x
j
i
p
i
)} which has the following defini-
tion: The state X
p
n
= x
j
n
p
n
, for the parent X
p
n
, is com-
patible with the state X
p
i
= x
j
i
p
i
, if according to the
expert’s mental model the state X
p
n
= x
j
n
p
n
is most
likely to coexist with the state X
p
i
= x
j
i
p
i
. Then
{Comp(X
p
i
= x
j
i
p
i
)} denotes the compatible parental
configuration where X
p
i
is in the state x
j
i
p
i
and the rest
of parents are in states compatible with X
p
i
= x
j
i
p
i
. And
then the WSA is used to generate the probabilities of
a child node’s CPT.
In literature, they frequently say that we need only
the 3
rd
term of equation n
o
5 as information from ex-
pert (Das, 2004). In reality, there is more information
to solicit from him: the relative weights for the parent
nodes (the 1
st
term) and the parental compatible con-
figurations (the 2
nd
one). Hence, the total number of
informations (NI) that we will need from the expert to
form the CPT will be:
NI = k +
k
∑
i=1
n
i
+
k
∑
i=1
(n
i
.(m − 1)) = k + m.
k
∑
i=1
n
i
(5)
where k the number of parent nodes.
This method presents two major issues when we
ask an expert to give us the compatible parental con-
figuration. The first issue is when the expert is con-
fronted to many compatible parental configuration for
a given parental state and he can’t select one of them.
To solve this problem, (Baker and Mendes, 2010) pro-
pose an extension for the WSA method by averaging
the probabilities of valid compatible parental configu-
rations that expert might select. The second issue is
when the expert is unable to give compatible parental
configuration since there are states of the nodes that
are not coherent with each other. To remedy this is-
sue, we propose a clustering approach detailed below.
4.2 Clustering Approach
The blocking situation is when the expert considers
that the compatible parental configuration of all pa-
rent nodes is irrelevant and he is unable to give all
the compatibilities as shown in the Figure 3. Indeed,
the expert explains that the nodes 2, 3, 4, 5 explai-
ned in Table 3, make sense from the point of view of
compatibility since they contribute to define, somew-
here the notion of cavitation (see the detail in the case
study section). That it is not objective to amalgamate
them with the nodes 1 and 6. This recurrent situa-
tion inspired us to propose the solution hereafter. We
propose to suggest to the expert, in elicitation phase,
to choose the most coherent group to define a com-
patibility. For this, he will propose a ”Cluster” (like
the one in Figure 4 grouping the nodes from 2 to 5)
whose notion of compatibility is plausible and insert
an intermediate node. In other words, we introduce
the Node Divorcing method (Fenton and Neil, 2012).
Subsequently, we can apply the WSA for this cluster
to obtain the CPT of the intermediate node. As for
the rest of nodes, they will be used for elicitation in-
dependently. Of course, it is possible that we have to
define several clusters. In this paper, only the case of a
single cluster will be treated to illustrate our method.
The Figure 2 illustrate a general example of a Bay-
esian network where the child node x
c
has a set of
parent nodes P from X
P
1
to X
P
k
. By applying our met-
hod, we obtain a cluster P
0
grouping the nodes X
P
6
,
X
P
7
and X
P
8
with an intermediate node noted x
Int
.
...
1 2 3 4 5 6
7
1 2 3 4 5 6
7
8
... ...
Figure 2: Bayesian network showing cluster Approach.
With our clustering approach, the total number of
informations (NI’) needed to elicit from the expert is
calculated with our equation 6 bellow:
NI
0
= k
0
+ m
0
.
∑
i∈P
0
n
i
+ (m −1).m
0
∏
i∈P−P
0
n
i
(6)
Where:
• k
0
: the number of parent nodes of the intermediate
node (cluster size), with k
0
∈ [2, k − 1].
• m
0
: the number of states of the intermediate node,
with m
0
∈
2,
k
0
∏
i=1
n
i
.
• P: set of the parent nodes index of the original
structure.
• P
0
: set of the parent nodes index of the intermedi-
ate node.
After presenting our method, we will develop it more
in the next section with case study and we detail our
results.
5 CASE STUDY
We relaunched our research work from a real case of
a petroleum company. Due to its importance in con-
tribution to the national economy and specifically in
the energy sector, the goal of this company is to de-
crease the waste of time and money. Thus, a sud-
den and instant stop in one of its production equip-
ment can cause enormous material damage with very
high maintenance costs, knowing that the acquisition
of new equipment will be too expensive. Since 2004,
the stopping frequency and maintenance interventi-
ons of the P-0701 pump have increased remarkably.
The cost of maintenance is too expensive and requires
qualified staff. However, they are uncertain to know
immediately the main causes for this judgment. In
short, they will waste time for the diagnosis, search
for causes and finally to repair them. In addition, if
the failure requires a spare part, it would be unavai-
lable on the market and must be ordered. To import
this piece, there will be a waste of time and the whole
process will be paralyzed. This will negatively in-
fluence the volume of production and subsequently a
large expense. Moreover, the pump played a signifi-
cant and critical role because it ensures the transfer
of gas from the field to its customers. So, if there is
a break or shutdown in the pump it will generate a
complaint and dissatisfaction of its customers.
To avoid these different consequences of stopping
the pump, it is useful to develop maintenance plans as
little as possible. To do this, we need a model with
a capacity of integrating technical and statistical data
of pump as well as knowledge expert. By the way, we
chose to model this problem by a model derived from
a probabilistic approach by the Bayesian networks to
estimate the causes that provide the shutdown of this
pump which will be used for decision making, rapid
diagnosis of the failure and prognostic in future. For
this, we made several meetings with the experts. We
used the method of 5 Why, the Ishikawa Diagram and
various questionnaires. The purpose was to build up
a body of knowledge and practices of maintenance
personnel that would produce strategies and plans for
maintenance operations of the pump system. The fin-
ding of this study was the important number of infor-
mation which is needed from experts to parameterize
the Bayesian structure successfully. It is at this point
that the usefulness of facilitation and reduction met-
hods for the elicitation of the model has become una-
voidable.
After validation with the experts, a small repre-
sentative part of the Bayesian structure is presented in
Figure 3. For reasons of simplification, we have re-
duced ourselves to the first level of causes of failure.
This allowed us to identify six main causes that can
lead to the sudden shutdown of the pump with two
possible states for each of them (Yes, No), (see Ta-
ble 3).
1 2 3 4 5 6
7
1 2 3 4 5 6
7
8
Figure 3: Part of Bayesian network of Pump P-0701 Shut-
down.
Table 3: Dictionary of nodes information.
Ref. Nodes
1 HUMAN FAULT
2 HYDRAULIC INSTABILITY
3 STEAM BUBBLE
4 TEMPERATURE
5 PRESSURE
6 PREMATURE WEAR
7 SHUTDOWN PUMP
Then, the expert must provide us the CPT of the
node 7. In this case, he must elicit, according to
the equation 2, NP = (2 − 1)
6
∏
i=1
n
i
= 64 probabili-
ties which is important and very hard to expert to give
us. In order to reduce the number of probabilities to
elicit and facilitate the task to the expert, we use the
WSA method detailed in previous section. Through
this method, we will go from 64 probabilities to eli-
cit to NI = 6 + 2 ×
6
∑
i=1
n
i
= 30 either a reduction of
53,12% which illustrates the linear growth of the in-
put information required by the method, compared to
the exponential growth of manual elicitation. Howe-
ver, when we asked the expert to give us the 12 pro-
bability distributions, he can’t do that. He is unable to
give all the compatibilities since there are states of the
nodes that are not coherent with each other. Indeed,
he can’t give the compatible parental configuration in-
cluding nodes 1 and 6 because they have no relation
with other nodes. To solve this problem, we introduce
our method and we ask the expert if he could propose
grouping or aggregation variables. So, he gives us
a grouping by introducing an intermediate node CA-
VITATION noted 8 with three states (Low, Average,
Important) as indicated in Figure 4.
Now, the expert is able to give us 2 + 2 + 2 + 2
probability distributions of parents nodes of the cavi-
tation node. For that, we gave him the Table 4 below
1 2 3 4 5 6
7
1 2 3 4 5 6
7
8
Figure 4: Part of Bayesian network of Pump P-0701 Shut-
down after divorcing node method.
as the tool to facilitate the task which present the com-
bination of all states of parent nodes and we asked him
just to check those who coexist the most with a given
state to make the compatible parental configurations.
Table 4: Compatible parental configurations.
2 3 4 5
Yes No Yes No Yes No Yes No
Yes x x x
2
No x x x
Yes x x x
3
No x x x
Yes x x x
4
No x x x
Yes x x x
5
No x x x
After that, we asked him to give us for each parent
nodes the relative weights w
1
, w
2
, w
3
et w
4
. For this,
we gave him a scale of weights (Figure 5 (b)) as a
second tool and we explain to him that the interval
widths for each state are 0,2 ; for example: the value
”High” is associated with the interval [0, 6 −0, 8], etc.
Figure 5: (a) Probability Scale given to the expert to de-
termine the conditional probabilities of the compatible pa-
rental configurations (b) Weights Scale which was given to
the expert to determine the relative weight for each parent
nodes.
Then, he will complete the 2× (3 − 1) probability
of each compatible parental configurations {Comp(2
= x)}, {Comp(3 = x)}, {Comp(4 = x)} and {Comp(5
= x)} while:
I
∑
y=L
P(8 = m|{Comp(X = x)}) = 1 (7)
where y present one state of the cavitation node (Low,
Average, Important), X = 2, 3, 4, 5 and x present one
state of the node used (Yes, No).
To facilitate the task of elicitation to the expert, we
provided him a scale of probabilities (Figure 5 (a)),
as the third tool, to give the conditional probabilities
corresponding to the compatible parental configura-
tions for every parental State. The following Tables
5 illustrate probability distribution of cavitation node
obtained from the expert related to compatibility sta-
tes Comp(2).
Table 5: Distribution of cavitation node related to compati-
ble parental configuration of node 2.
Probability Distribution over CAVITATION Yes No
P(8 = Low|{Comp(2)}) 0,05 0,8
P(8 = Average|{Comp(2)}) 0,35 0,15
P(8 = Important|{Comp(2)}) 0,6 0,05
Taking these distributions and the weights as in-
put, the WSA would be able to calculate all the
(3 − 1)
4
∏
i=1
n
i
= 32 distributions required to build the
CPT of cavitation node.
Now, we ask the expert to fill us the CPT of node 7
which has, after applying the node divorcing method,
tree parent nodes instead of six. In this case, he will
elicit just (2 − 1)
3
∏
i=1
n
i
= 12 probabilities.
Then, we summarize in Table 6 the elicitation in-
formation needed for different methods.
Table 6: Elicitation information needed for different met-
hods.
Method NI To elicit Percentage of information reduced
Raw 64
WSA 30 53,12 %
Cluster-WSA 40 37,5%
We note that the rate of reduction number of in-
formation to elicit is not very important when we use
our method. However, if we take more complicated
example with several possible states, the gap will be
important. To better understand and see the results,
we made two curves showing the three methods by
varying cluster size k
0
.
For the first curve (see Figure 6), it’s our case
study with m
0
= 3. We observe that as the size clus-
ter increases, the number of information with Cluster-
WSA decreases between 2 and 4 until it reaches 40
as a minimum value of information needed. Beyond
! ! ! !
!
Figure 6: Elicitation information needed according to clus-
ter size with 3 states of intermediate node.
that it increases. We concluded that our method is bet-
ter than the Raw method whereas it is less good with
WSA concerning the number of information to elicit.
For the second curve (see Figure 7), we choose
m
0
= 2 the minimum number of possible states which
can be given for the intermediate node. This time,
! ! ! !
"
#
Figure 7: Elicitation information needed according to clus-
ter size with 2 states of intermediate node.
we notice that the number of information needed with
cluster-WSA has decreased more than with the WSA
method where k
0
= [4, 5]. In this case, our cluster-
approach is better than the two other methods.
To summarize, it is true that cluster-WSA does not
always give information in some cases compared to
WSA. However, through the use of our method, we
have remedied the scientific issue encountered.
6 CONCLUSION
Eliciting probability parameters for CPT is one of the
important problems for BN. There are many methods
of elicitation in the literature that facilitate and reduce
the task of elicitation. The most generic is the WSA
which was developed in this paper. This method is
easy to apply and it makes the number of CPT’s asses-
sments linear instead of exponential. However, it pre-
sents two major issues when the expert must give the
compatible parental configuration. The first is where
the expert cannot select a single compatible parental
configuration for a given state. This issue was solved
in the literature. By contrast, the second isn’t. The
problem is when expert still unable to give compatible
parental configuration for each state with other states
of other parents. As a solution for this issue, we pro-
posed, in this paper, a cluster approach by including
a intermediate node. We applied the WSA on a gene-
rated cluster and we used Raw method of elicitation
for the rest of nodes. Then, we applied our method
on a part of BN of our case study. To facilitate the
application of WSA, we have combine various diffe-
rent concepts. Among which we elaborate a table for
compatible parental configurations which is easy to
complete by the expert. In addition, we use a scale
of probabilities and a scale of weights to facilitate the
task of elicitation to him. In addition, we developed
two practices equations to calculate the number of in-
formation needed from expert one for the WSA and
the other for our method. After that, we compared in-
formation number needed obtained from our method
with WSA and the Raw method by varying the cluster
size and states number of intermediate node. Through
using our method, the information number to elicit
is less important than that obtained by the Raw met-
hod. But this is not always the case compared with
the WSA. This comparison allows us to admit that
our clustering method is better than the Raw method
but it’s still not good sometimes compared to WSA
knowing that it has remedied the scientific issue en-
countered.
It is true that our clustering approach reduces the
number of information to be elicited in order to deve-
lop our CPT and remedied the scientific issue encoun-
tered with WSA, nevertheless like other methods, it
has some limits. In our next research, we will try to
find them and to compare our method with other exis-
ting methods. Moreover, in this paper, we present our
method to remedy the problem of compatible paren-
tal configuration and we have treated a single cluster.
As a prospect, we may wonder what will become our
method and results if we have several clusters.
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