Probabilistic Cognitive Maps
Semantics of a Cognitive Map when the Values are Assumed to be Probabilities
Aymeric Le Dorze
1
, B
´
eatrice Duval
1
, Laurent Garcia
1
, David Genest
1
, Philippe Leray
2
and St
´
ephane Loiseau
1
1
Laboratoire d’
´
Etude et de Recherche en Informatique d’Angers, Universit
´
e d’Angers,
2 Boulevard Lavoisier, 49045, Angers Cedex 01, France
2
Laboratoire d’Informatique de Nantes Atlantique,
´
Ecole Polytechnique, Universit
´
e de Nantes,
La Chantrerie - rue Christian Pauc, 44306, Nantes Cedex 3, France
Keywords:
Cognitive Map, Probabilities, Causality, Bayesian Network.
Abstract:
Cognitive maps are a knowledge representation model that describes influences between concepts by a graph,
where each influence is quantified by a value. The values are generally not formally defined. In this paper,
we introduce a new cognitive map model, the probabilistic cognitive maps. In such maps, the values of the
influences are interpreted as probability values. We define formally the semantics of this model. We also
provide an operation to compute the global influence of a concept on any other one, called the probabilistic
propagated influence. To show that our model is valid, we propose a procedure to represent a probabilistic
cognitive map as a Bayesian network. This new model strengthens cognitive maps by giving them strong
semantics. Moreover, it acts as a bridge between cognitive maps and Bayesian networks.
1 INTRODUCTION
Graphical models for knowledge representation help
to easily organize and understand information. A cog-
nitive map (Axelrod, 1976) is a graph that represents
influences between concepts. A concept is a short
textual description of a part of the real world such
as an action or an event and is represented by a la-
belled node in the graph. An influence is an arc be-
tween these concepts. A cognitive map provides an
easy visual communication medium for humans, es-
pecially for the analysis of a complex system. It can
be used for instance to take a decision in a brainstorm-
ing meeting. These maps are used in several domains
such as biology (Tolman, 1948), ecology (Celik et al.,
2005), or politics (Levi and Tetlock, 1980).
In a cognitive map, each influence is labelled with
a value that quantifies it. This value describes the
strength of the influence. It belongs to a previously
defined set, called a value set. A cognitive map can
be defined on several kinds of value sets. These
value sets can be sets of symbolic values such as
{+, −} (Axelrod, 1976) or {none, some, much, a
lot } (Dickerson and Bart, 1994; Zhou et al., 2003), or
an interval of numeric values such as [−1; 1] (Kosko,
1986; Satur and Liu, 1999). Thanks to these values,
we are able to compute the global influence of any
concept of the map on any other one. Such an oper-
ation is called the propagated influence. To compute
this propagated influence, the values of the influences
that compose the paths linking the two concepts are
aggregated according to their semantics.
The main advantage of cognitive maps is that
they are simple to use; people who are not familiar
with formal frameworks need this simplicity. Conse-
quently, the semantics of the values are sometimes not
clearly defined. The drawback is that it is often hard
to interpret the real meaning of the values associated
to the influences and to verify the soundness of the
computed propagated influence.
Some approaches exist to define formally the se-
mantics of cognitive maps. The fuzzy cognitive
maps links the cognitive maps to the fuzzy set frame-
work (Kosko, 1986; Aguilar, 2005). They consider
that the concepts are fuzzy sets and that the values
represent the degrees of causality between these con-
cepts. These maps are generally easy to use but the in-
ference is sometimes quite obscure for a layman since
fuzzy sets are not a very popular framework.
There exist other knowledge representation mod-
els that represent both a graph and values associated
to a strong semantic. The graphical structure of a cog-
52
Le Dorze A., Duval B., Garcia L., Genest D., Leray P. and Loiseau S..
Probabilistic Cognitive Maps - Semantics of a Cognitive Map when the Values are Assumed to be Probabilities.
DOI: 10.5220/0004757200520062
In Proceedings of the 6th International Conference on Agents and Artificial Intelligence (ICAART-2014), pages 52-62
ISBN: 978-989-758-015-4
Copyright
c
2014 SCITEPRESS (Science and Technology Publications, Lda.)
nitive map and the values given by a concept influenc-
ing another one remind us of the Bayesian network
framework (Pearl, 1988; Pearl, 2009). Bayesian net-
works are graphical models for knowledge represen-
tation that express dependency relations between vari-
ables. These relations are quantified with conditional
probabilities. They are more expressive than cogni-
tive maps but their building and their use are more
complex. It is then interesting to improve the formal
aspect of cognitive maps when dealing with values as-
sumed to be probabilities since probabilities are gen-
erally a popular framework. Such a model would keep
the simplicity of cognitive maps while tending to be
as formal as Bayesian networks.
This paper introduces a new cognitive map model,
the probabilistic cognitive maps. This model keeps
the simplicity of cognitive maps while improving the
formal representation of the values by providing a
probabilistic interpretation for the influence values.
Such an interpretation is formal enough without be-
ing restrictive to users but needs to adapt the seman-
tics of the concepts and the influences. For the same
reason, the propagated influence needs to be rede-
fined to fit the semantics. To show the validity of
our model, we propose a procedure to represent a
cognitive map as a Bayesian network and show that
the propagated influence in the probabilistic cogni-
tive map corresponds to a specific probability in the
Bayesian network. The studied Bayesian network
model is the causal Bayesian network model (Pearl,
2009) because, as shown in this paper, it is more
closely related to cognitive maps.
There exist other works that ties cognitive maps to
probabilities. For example, (Song et al., 2006) defines
the fuzzy probabilistic cognitive map model, which is
based on the fuzzy cognitive map model. However,
in this model, the probabilities are only expressed
on the concepts as it is used to compute whether a
concept can influence other concepts or cannot. The
probabilistic cognitive map model that we define must
not be confused with the Incident Response Proba-
bilistic Cognitive Map model (IRPCM) (Krich
`
ene and
Boudriga, 2008). In this model, the links between
the concepts are not necessarily causal, therefore what
they call a ”cognitive map” is not the same model as
the one we define here. IRPCM is mostly used for
diagnosis whereas our model proposes a framework
that studies influences between concepts.
In this article, we present in section 2 the cogni-
tive map model and a simple introduction to Bayesian
networks. In section 3, we define the probabilistic
cognitive map model as well as the semantics of the
concepts and the influences and the propagated influ-
ence in this model. In section 4, we justify our model
by encoding a cognitive map into a causal Bayesian
network.
2 STATE OF THE ART
In this section, we first present the cognitive map
model in section 2.1. Then, we present the Bayesian
network model in section 2.2. Finally, we present the
causal Bayesian network model in section 2.3.
2.1 Cognitive Maps
A cognitive map is a knowledge representation model
that represents influences between concepts with a
graph. An influence is a causal relation between two
concepts labelled with a value that quantifies it. It ex-
presses how much a concept influences another one
regardless of the other concepts. This value belongs
to a predefined set, called the value set.
S
0.9
R
0.8
0.8
G
0.6
N
−0.1
P
S Sprinkler
R Rain
G Wetness of my garden
N Wetness of my neighbour’s garden
P Health of my plants
Figure 1: CM1, a cognitive map defined on the value set
[−1;1].
Definition 1 (Cognitive Map). Let C be a concept set
and I a value set. A cognitive map CM defined on I is
a directed graph CM = (C, A, label) where:
• the concepts of C are the nodes of the graph;
• A ⊆ C ×C is a set of arcs, called influences;
• label : A → I is a function labelling each influence
with a value of I.
Example 1. The cognitive map CM1 (figure 1) repre-
sents the influences of some concepts on the health of
my plants. It is defined on the value set [−1;1]. An in-
fluence between two concepts labelled with a positive
value means that the first concept positively influences
the second one. A negative value means on the con-
trary that the first concept negatively influences the
second one. A value of 1 means that the influence
is total. A value of 0 means that there is no direct
influence between two concepts whereas the absence
of an influence between two concepts means that the
ProbabilisticCognitiveMaps-SemanticsofaCognitiveMapwhentheValuesareAssumedtobeProbabilities
53
builder of the map does not know if there is such a
relation between these concepts.
If we consider the concepts R and G, the rain in-
fluences the humidity of my garden by 0.8. On the
contrary, if we consider the concepts N and P, the hu-
midity of my neighbour’s garden influences the health
of my plants by −0.1 because his growing trees shade
my garden.
Thanks to the influence values, the global influ-
ence of a concept on another one can be computed.
This global influence is called the propagated influ-
ence and is computed by aggregating the values on
the influences that belong to any path linking these
two concepts. Many algorithms to compute the prop-
agated influence exist. We will present only the most
common one for the value set [−1; 1] (Chauvin et al.,
2013). It is composed of three steps.
The first step is to list the different paths that link
the first concept to the second one. Since a cogni-
tive map may be cyclic, there is potentially an infi-
nite number of paths between the two concepts. To
avoid an infinite computation, only the most mean-
ingful paths are considered, which are the paths that
does not contain any cycle. Indeed, if a path con-
tains a cycle, it means that a concept influences it-
self. Because the effect of this influence cannot have
immediate consequences, it occurs in fact at a future
time frame. Therefore, since the influences of a path
should belong to the same time frame, the paths that
contain a cycle are not considered. A path that con-
tains no cycle is called a minimal path.
The second step is to compute the influence value
that each of these paths brings to the second concept.
This influence value is called the propagated influ-
ence on a path and is denoted by I P . To compute it,
the influence values of the said path are simply multi-
plied together.
Finally, the third step is to aggregate the propa-
gated influences on every minimal path that links the
first concept to the second one with an average. The
propagated influence I of a concept on another one is
thus defined as the sum of the propagated influences
on every minimal path between the two concepts di-
vided by the number of minimal paths.
Definition 2 (Propagated Influence). Let c
1
and c
2
be
two concepts.
1. An influence path P from c
1
to c
2
is a sequence
of length k ≥ 1 of influences (u
i
, u
i+1
) ∈ A with
i ∈ [0;k − 1] such that u
0
= c
1
and u
k
= c
2
. P
is said minimal iff ∀i, j ∈ [0; k − 1], i 6= j ⇒ u
i
6=
u
j
∧u
i+1
6= u
j+1
; we denote by P
c
1
,c
2
the set of all
minimal paths from c
1
to c
2
.
2. The propagated influence on P is:
I P (P) =
k−1
∏
i=0
label
(u
i
, u
i+1
)
3. The propagated influence of c
1
on c
2
is:
I (c
1
, c
2
) =
(
0 if P
c
1
,c
2
=
/
0
1
|P
c
1
,c
2
|
×
∑
P∈P
c
1
,c
2
I P (P) otherwise
Example 2. In CM1, we want to compute the propa-
gated influence of R on P.
1. there are two minimal paths between R and P:
p
1
= {R → G → P} and p
2
= {R → N → P};
2. the propagated influences on p
1
and p
2
are:
I P (p
1
) = 0.8 × 0.6 = 0.48 and I P (p
2
) = 0.8 ×
−0.1 = −0.08;
3. the propagated influence of R on P is: I (R, P) =
1
2
×
I P (p
1
) + I P (p
2
)
=
1
2
× (0.48 − 0.08) =
0.2.
2.2 Bayesian Networks
Bayesian networks (Pearl, 1988; Pearl, 2009) are
graphical models that represent probabilistic depen-
dency relations between discrete variables as condi-
tional probabilities. Each variable takes its value from
many predefined states. In such a graph, each variable
is assimilated to a node and an arc represents a prob-
abilistic dependency relation between two variables.
This graph is acyclic. Each variable is associated to a
table of conditional probabilities. Each entry of this
table provides the probability that a variable has some
value given the state of each parent of this variable in
the graph.
A Bayesian network allows to compute the prob-
abilities of the states of the variables according to the
observation of some other variables in the network.
The structure of the graph is used to simplify the
computations by using the independence relations be-
tween the variables. However, these computations are
generally NP-complete (Chickering, 1996).
S
R
G
N
S I let my sprinkler on last night.
R It rained last night.
G The grass of my garden is wet.
N The grass of my neighbour’s garden is wet.
Figure 2: The Bayesian network BN1.
Example 3. The Bayesian network BN1 (figure 2)
represents dependency relations between variables
ICAART2014-InternationalConferenceonAgentsandArtificialIntelligence
54
P
S 0.4
S 0.6
P
R 0.4
R 0.6
P SR SR SR SR
G 1 1 1 0
G 0 0 0 1
P R R
N 1 0
N 0 1
Figure 3: The probability tables of the variables of the
Bayesian network BN1.
related to the humidity of my garden. These variables
are binary events. We denote the state A = > by A
and A = ⊥ by A for any event A.
Each node is associated to a probability table (fig-
ure 3). The first row of the first table means that
the probability that I let my sprinkler on last night
is P(S) = 0.4. The values in the table of the variable
G means that I am sure that my garden is wet either
if I let my sprinkler on last night, or if it rained last
night, or both. Otherwise, I am sure that my garden is
not wet.
From this network, some information can be de-
duced, like the probability of the states of each node
or the independence of two nodes. We can also com-
pute conditional probabilities.
For example, as I am leaving my home, I notice
that the grass of my garden is wet. The grass can only
be wetted by the rain or my sprinkler. So, I ask myself
if I have let my sprinkler on. Thanks to this network,
we compute P(S|G) = 0.625. This value is greater
than P(S). This means that knowing that my garden
is wet increases the probability that I let my sprinkler
on. However, we also compute P(R|G) = 0.625. Thus,
we are unable to know what has wetted my garden
between my sprinkler and the rain as these events are
equiprobable given that my garden is wet.
Then, I notice that the grass of neighbour’s gar-
den is not wet. If it rained last night, then both
our gardens should be wet. We need so to compute
the probability that my sprinkler is on given that my
grass is wet, contrary to my neighbour’s. We com-
pute P(S|GN) = 1. Thus, I am now sure that I let my
sprinkler on.
2.3 Causal Bayesian Networks
The causal Bayesian network model (Pearl, 2009)
extends the classical Bayesian network model. The
main difference between these two models is the fact
that the arcs of a Bayesian network can represent any
kind of probabilistic dependency whereas they have
to be causal in a causal Bayesian network. Con-
trary to classical Bayesian networks, causal Bayesian
networks also distinguishes observation and interven-
tion. When an observation is made on a variable, the
information is propagated to the nodes linked to this
variable regardless of the direction of the arcs. When
an intervention is made on a variable, the information
is propagated only to its children, following the di-
rection of the arcs. Thus, with intervention, only the
descendants of the variable are influenced by it.
For example, if I observe that my garden is wet
and I want to compute the probability that it rained
last night, I have to compute P(R|G), as discussed
earlier. That kind of reasoning can be both deduc-
tive and abductive (Charniak and McDermott, 1985).
Now, if I make my garden wet, I intervene on the
humidity of my garden. To represent that interven-
tion, the causal Bayesian network model defines a
new operator, called do(·) (Pearl, 2009). Here, if I
want to compute the probability that it rained given
the fact that I made my garden wet, I have to compute
P(R|do(G)). Applying do(G) is so equivalent to re-
move the arcs ending on G in the Bayesian network
and separate it from its parents (Spirtes et al., 2001).
Intuitively, the fact that I made my garden wet has no
consequence whatsoever on the fact that it rained. So,
P(R|do(G)) = P(R). That kind of reasoning is strictly
deductive and only affects the descendants of G.
3 THE PROBABILISTIC
COGNITIVE MAP MODEL
In this section, we define the probabilistic cognitive
map model. In such a cognitive map, the influence
values are interpreted as probability values. The se-
mantics of the concepts and the influences must be
defined according to this interpretation. For the same
reason, the propagated influence of a concept on an-
other one must be redefined according to these seman-
tics.
We first clarify what kind of information a concept
and an influence represent in section 3.1. Then, we
define how to compute the propagated influence of a
concept on another one in section 3.2. We call such
a propagated influence the probabilistic propagated
influence.
3.1 Semantics of the Concepts and the
Influences
The simple cognitive map of figure 4 allows us to ex-
plain easily the idea behind the notion of influence in
terms of probabilities. Note that in the general case,
the relationships between the influences, the values
and the probabilities are more complex but this ba-
ProbabilisticCognitiveMaps-SemanticsofaCognitiveMapwhentheValuesareAssumedtobeProbabilities
55
sic example helps to get the basic idea behind our
approach. The simple map only represents a single
influence from a concept A to a concept B with an in-
fluence value α. Such a map means that A influences
B at a level α. Since α is a probability, the concepts A
and B must be associated to random variables.
A
α
//
B
Figure 4: A simple cognitive map.
A random variable is generally associated to sev-
eral disjoint values covering the set of its possible
states. We would like this set to be as small as possible
and to be the same for every variable associated to a
concept, to keep the simplicity of the model. These
values need to represent an information of the real
world.
In a cognitive map, a concept is often associated to
a piece of information of the real world which is quan-
tifiable. For example, if we consider the concept S in
example 3, it can be seen as the strength of the sprin-
kler or as the quantity of water it delivers. We define
the possible values of the random variable associated
to the concept using this quantity. However, we can-
not use directly the possible values of this quantity
since it may be a continuous scale.
In order to have the same set of values for ev-
ery random variable, we define two values, inspired
by (Cheah et al., 2007). The value + means that the
concept is increasing. The value − means that the
concept is decreasing.
Example 4. We consider the concept S that repre-
sents a sprinkler from the example 1. The quantity
associated to S is the quantity of water that the sprin-
kler is delivering.
We define the random variable X
S
associated to S. The
increase state X
S
=+ means that S is increasing, that
is the sprinkler is delivering more and more water.
The decrease state X
S
=− means that S is decreasing,
that is the sprinkler is delivering less and less water.
Note that we do not provide a state that repre-
sents the fact that a concept is stagnating. This im-
plies that the quantity associated to the concept can-
not remain unchanged and has to either increase or
decrease. However, we consider that this should not
have strong consequences as cognitive maps aim to
study influences between concepts. Thus, we are not
interested to know if a concept stagnates but rather if
a concept influences another one.
Note also that in (Cheah et al., 2007), the state
X
S
=+ means that the causal effect of S is positive
whereas X
S
=− means the effect is negative. This
representation is close to ours but the semantics of the
causal effect is stronger with our approach.
Now that the states of the random variables as-
sociated to the concepts are defined, we have to ap-
ply a probability law on these states. To compute
the probabilistic propagated influence, we need the a
priori probability of the states of every random vari-
able of the map. The a priori probability of a state is
given when we have no information about the states
of any concept. Since there is no information in a
cognitive map providing the a priori probability of
any state of any concept, we assume that the states of
every random variable of the map are equiprobable.
Since the random variable associated to each concept
has only two states, for every concept A of the map,
P(X
A
=+) = P(X
A
=−) = 0.5.
We focus now on the semantics of the influence
values. To evaluate the influence of a concept on
another one, the idea is to study how the influenced
concept reacts relatively to the different states of the
influencing concept. In our case, this leads to study
the probabilities of the states of the influenced con-
cept given that the influencing concept is increasing
or decreasing. Therefore, if we consider the simple
map from figure 4, the influence between A and B
is linked to the probabilities of X
B
when X
A
=+ and
when X
A
=−. The value α of an influence should rep-
resent how the influenced concept reacts and is thus
tied to these conditional probabilities.
A has two ways to influence B: either when A is
increasing or when A is decreasing. Thus, the influ-
ence should have two values: one for the state X
A
=+,
and one for the state X
A
=−. (Sedki and Bonneau de
Beaufort, 2012) labels each influence of a cognitive
map with two values. However, we want only one
value for each influence in the cognitive map, in or-
der to keep the simplicity of the model. Therefore,
we need to express a relation between the two val-
ues. According to (Kosko, 1986), we assume that, an
influence being a causal relation, the effect of the in-
crease of A on the increase of B equals the effect of the
decrease of A on the decrease of B. Thus, the proba-
bility of X
B
when X
A
=+ should be the complement
of the probability of X
B
when X
A
=−. In our model,
we consider that the influence value α represents the
influence of A on B when they are both increasing.
Giving a value α to the direct influence between A
and B would lead to answer questions such as ”Given
that A is increasing, how the probability that B is in-
creasing is modified?”. The influence value α quan-
tifies the modification of the a priori probability of B
caused by A, in other words, the difference between
the conditional probability of B given that A is in-
creasing and the a priori probability of B. Thus, α
is linked to the difference between P(X
B
=+|X
A
=+)
and P(X
B
=+).
ICAART2014-InternationalConferenceonAgentsandArtificialIntelligence
56
This relation between the notion of influence and
a conditional probability has consequences on the
structure of the cognitive map. Indeed, to compute
the global influence of a concept on another one, we
aggregate influences. Thus, when we compute the
global influence, we manipulate in fact conditional
probabilities. Therefore, the global influence of a con-
cept on itself is linked to the conditional probability
of a variable given that variable. In such a case, the
value of the conditional probability must check cer-
tain properties: for example, it has to be equal to
either 0 or 1 according to the different values of the
variable. Thus, if there are influences that link a con-
cept to itself, the values of these influences should re-
spect this property. As we consider this constraint too
strong for the designer of a cognitive map, we forbid
cycles in a probabilistic cognitive map.
Now, we express formally the link between
α and the difference between P(X
B
=+|X
A
=+)
and P(X
B
=+). Since P(X
B
=+) = 0.5 and
P(X
B
=+|X
A
=+) is a probability value and must be-
long to [0; 1], α should belong to [−0.5; 0.5]. How-
ever, in the cognitive map of the example 1, it is ob-
viously not the case as this map is defined on [−1;1].
The idea is to convert α into a value of [−0.5; 0.5].
Therefore, a conversion function F must be defined
such that whatever the value set I the cognitive map
is defined on, its values are converted into values of
[−0.5, 0.5]. Moreover, a reverse conversion function
F
-1
is defined to get back an influence value that be-
longs to I when the computation of the propagated
influence is done. This reverse conversion function
is defined such that F
-1
F (α)
= α. If the conver-
sion function is bijective, then the reverse conversion
function is simply its reciprocal function.
The conversion function allows us to say that we
have F (α) = P(X
B
=+|X
A
=+) − P(X
B
=+). Note
that this relation is more complex when B has more
than one parent.
Example 5. Since the cognitive map CM1 is de-
fined on [−1;1], we define the conversion function
F : [−1; 1] → [−0.5;0.5] as F (α) =
α
2
. We define
the reverse conversion function F
-1
: [−0.5;0.5] →
[−1;1] as F
-1
(α) = α × 2.
Now that the semantics of a direct influence are
established, we define how to combine influences to
compute the propagated influence in a probabilistic
cognitive map.
3.2 Probabilistic Propagated Influence
We call the operation of propagated influence in a
probabilistic cognitive map the probabilistic propa-
gated influence. We consider that such an influence
should take its values in the same value set as the one
the cognitive map is defined on. However, we have
stated that the value of a direct influence is linked
to the difference between a conditional probability
and an a priori probability and that this difference be-
longs to [−0.5; 0.5]. The propagated influence being
the combination of many direct influences, its value
should also belong to [−0.5;0.5]. Before computing
the probabilistic propagated influence, we compute
what we call the partial probabilistic propagated in-
fluence I
P
0
that represents this difference. As it takes
its values in [−0.5; 0.5], we use the reverse conver-
sion function to compute the probabilistic propagated
influence and get back a value of the original value
set.
To compute the partial probabilistic propagated
influence of a concept on another one, we follow the
same procedure as for the propagated influence de-
scribed in definition 2. First, we list the paths be-
tween the two concepts. Then we compute the influ-
ence value of each path. Finally, we aggregate these
influence values.
Since a probabilistic cognitive map is acyclic, the
set of paths between two concepts is necessarily finite.
We need then to compute the influence value of
each one of these paths. The probabilistic propagated
influence on a path I P
P
represents the influence value
of the said path. To compute this value, we cannot
simply multiply the converted values in the same way
we did for the values of [−1;1] in the previous section
as the result of such a product would belong to some-
thing like [−(0.5
n
);0.5
n
]. A better way to aggregate
the values is to multiply the converted values by 2 be-
fore the product and then divides the final result by 2.
Thus, we get a value that belongs to [−0.5;0.5].
Definition 3 (Probabilistic propagated influence on a
path). Let F be a conversion function. Let P be a path
of length k and made of influences (u
i
, u
i+1
) between
two concepts of CM. The probabilistic propagated in-
fluence on P is:
I P
P
(P) =
1
2
×
k−1
∏
i=0
2 × F
label
(u
i
, u
i+1
)
Example 6. We consider the path p
1
= R → G → P
in CM6 (example 2). We use the conversion function
defined in example 5. The probabilistic propagated
influence on p
1
is:
I P
P
(p
1
) =
1
2
×
2 × F (0.6)
×
2 × F (0.8)
= 0.24
To compute the probabilistic propagated influ-
ence, we aggregate the values of the probabilistic
propagated influences on the paths between two con-
cepts. This aggregation is also different from the one
ProbabilisticCognitiveMaps-SemanticsofaCognitiveMapwhentheValuesareAssumedtobeProbabilities
57
defined in the previous section. We need to weight
each path before the aggregation. This weight is
called the part of a path.
Let us consider that we are computing the proba-
bilistic propagated influence of a concept A on a con-
cept B. To compute the parts of the paths between A
and B, we follow them backwards, starting from the
concept B. We consider that each parent of B is given
the same weight. Then, for each one of these parents,
we share equally the previously given weight between
its own parents and so on. This recursion ends either
on A or on a root concept. If we end on A, then the
current weight is the part of the path that we followed,
starting from B. If we end on root concept other than
A, then we did not follow a path between A and B.
A graphical representation of this computation is
shown on figure 5. Let us consider for example the
leftmost path. Starting from B, we give to the parent
of B on this path a weight of
1
2
because B has 2 par-
ents. Then, we do the same operation for this parent
and, as it has 3 parents, the path has now a part of
1
2
×
1
3
=
1
6
. We do it again for the last arc to end on
A and to get a part of
1
12
. Since A is the influencing
concept, the part of this path is
1
12
.
◦
A
xx
&&
◦
1
12
33
◦
◦
1
4
VV
◦
◦
1
6
UU
◦
1
4
gg
◦
B
1
2
UU
1
2
II
Figure 5: Computation of the parts of two different paths
from A to B.
To sum up, the part of a path between A and B is
1 divided by the number of parents of each concept
crossed by this path, except A.
Definition 4 (Part of a path). Let P be a path of length
k and made of influences (u
i
, u
i+1
) between two con-
cepts of CM. Let C (c) denote the parents of any con-
cept c. The part of P is:
part(P) =
k
∏
i=1
1
|C (u
i
)|
Example 7. We consider again the path p
1
= R →
G → P from example 2. The part of p
1
is:
part(p
1
) =
1
|C (G)|
×
1
|C (P)|
=
1
2
×
1
2
=
1
4
Using the part and the probabilistic propagated in-
fluence on a path, we are able to compute the partial
probabilistic propagated influence of a concept on an-
other one. It is defined as the sum of the products of
the part and the probabilistic propagated influence on
each path between the two concepts. With such a def-
inition, when there is no path between two concepts,
the probabilistic propagated influence is 0, which is
what we would expect since there is no way any of
these concepts may influence the other one.
However, there is an exception to this definition
when we want to compute the probabilistic propa-
gated influence of a concept on itself. Since, for any
random variable X and any one of its possible values
x, we have P(X =x|X =x) = 1, we should have, for
any concept A, P(X
A
=+|X
A
=+) = 1. Since we de-
fined the partial probabilistic propagated influence of
a concept on another one as the difference between
a conditional probability and the a priori probability,
the partial probabilistic propagated influence of a con-
cept on itself should be 0.5.
Definition 5 (Partial probabilistic propagated influ-
ence). Let F be a conversion function. Let c
1
and c
2
be two concepts. The partial probabilistic propagated
influence of c
1
on c
2
is:
I
P
0
(c
1
, c
2
) =
(
0.5 if c
1
= c
2
∑
P∈P
c
1
,c
2
part(P) × I P
P
(P) otherwise
Example 8. We want to compute the partial proba-
bilistic propagated influence of R on P in CM1. We
already stated in exemple 2 that there is two paths be-
tween R and P: p
1
= R → G → P and p
2
= R → N →
P. We have also already computed I P
P
(p
1
) = 0.24
and part(p
1
) =
1
4
in examples 6 and 7. We compute
in the same way I P
P
(p
2
) = −0.04 and part(p
2
) =
1
2
.
The partial probabilistic propagated influence of R on
P is:
I
P
0
(R, P) = part(p
1
)×I P
P
(p
1
) + part(p
2
)×I P
P
(p
2
)
=
1
4
× 0.24 +
1
2
× −0.04 = 0.04
The partial probabilistic propagated influence of
N on S is I
P
0
(N, S) = 0, as there is no path linking the
two concepts.
The partial probabilistic propagated influence of
S on itself is I
P
0
(S, S) = 0.5.
We said earlier that the probabilistic propagated
influence is defined as the value of the partial proba-
bilistic propagated influence converted using the re-
verse conversion function. Looking closely at the
definition of the partial probabilistic propagated in-
fluence, we notice that this definition looks like a
weighted average of the probabilistic propagated in-
fluence on the paths. The weights are given by the
ICAART2014-InternationalConferenceonAgentsandArtificialIntelligence
58
respective parts of these paths. However, the sum of
these weights does not equal 1. Normalizing the par-
tial probabilistic propagated influence by the sum of
the parts of the paths before converting the value has
two advantages. First, we compute a real weighted
average. Second, it ensures that, if two concepts are
linked by a single direct influence, the probabilistic
propagated influence of the first concept on the sec-
ond one equals the value of the direct influence.
After this normalization is done, we can convert
the value using the reverse conversion function to get
our probabilistic propagated influence. Note that the
normalization cannot be done when there is no path
between the two concepts as we cannot divide by the
sum of the parts of the paths, which is 0. In that case,
we simply convert the partial probabilistic propagated
influence without any normalization.
Definition 6 (Probabilistic propagated influence). Let
F be a conversion function and F
-1
be its reverse
conversion function. Let c
1
and c
2
be two concepts.
The probabilistic propagated influence of c
1
on c
2
is:
I
P
(c
1
, c
2
) =
F
-1
I
P
0
(c
1
, c
2
)
if P
c
1
,c
2
=
/
0
F
-1
I
P
0
(c
1
,c
2
)
∑
P∈P
c
1
,c
2
part(P)
!
otherwise
Example 9. As in example 8, we compute this time
the propagated influence between R and P. We use
the reverse conversion function defined in example 5.
The probabilistic propagated influence of R on P is:
I
P
(R, P) = F
-1
I
P
0
(R,P)
part(p
1
)+part(p
2
)
=
0.04
1
4
+
1
2
× 2 = 0.1067
As there is no path between N and S, the prob-
abilistic propagated influence is 0 and for the same
reason, the probabilistic propagated influence of S on
itself is 1.
4 RELATIONS WITH THE
BAYESIAN NETWORK MODEL
In order to prove the validity of the probabilistic cog-
nitive map model and the definition of the proba-
bilistic propagated influence associated to it, we de-
fine a procedure to encode any probabilistic cognitive
map into a Bayesian network. We demonstrate also
that, in such a cognitive map, the computation of the
probabilistic propagated influence equals the compu-
tation of a specific conditional probability in the re-
lated Bayesian network.
We give first the idea of the encoding in sec-
tion 4.1. We then show more clearly the relation
between the probabilistic propagated influence and
a conditional probability in the associated Bayesian
network in section 4.2.
4.1 Encoding a Cognitive Map as a
Bayesian Network
The Bayesian network is built from the cognitive
map such that each node of the cognitive map (con-
cept) is encoded as a node in the Bayesian network.
Each influence between two concepts of the map is
also encoded as an arc between the two nodes in the
Bayesian network that represent these concepts. So,
the Bayesian network has the same graphical structure
as the cognitive map. Thus, we give the same name to
the cognitive map nodes and to the Bayesian network
nodes.
Since both the cognitive map and the Bayesian
network have the same structure and since a Bayesian
network is acyclic, we have also to be sure that the
map is acyclic. To remove the cycles of a cogni-
tive map, (Nadkarni and Shenoy, 2001; Nadkarni and
Shenoy, 2004) describe how to obtain a map structure
suitable for a Bayesian network. One way to prevent
cycles is to discuss with the map designer to explain
what is the meaning of the links to avoid redundancy
or inconsistency. Another way is to disaggregate a
concept of the cycle into two time frames. That is
why we consider only acyclic cognitive maps in this
paper.
Each node of the Bayesian network is associated
to a random variable that corresponds to the random
variable the concept of the cognitive map is associated
to. The probability table associated to each variable is
computed from the values of the influences that end to
the concept associated to this variable in the cognitive
map.
We consider first the nodes that have no parent.
With such nodes, the only probability values that we
have to provide are a priori probabilities. We already
know this values as we stated earlier that the different
states of a concept are equiprobable.
Example 10. The probability table of the node S from
example 1 is:
P
X
S
=+ 0.5
X
S
=− 0.5
For the other nodes that have many parents such as
shown in figure 6, we have to provide the conditional
probabilities for every possible state of their parents.
Thus, we have to merge the values from the arcs that
end to one of these nodes to express these probabili-
ties. There are several methods to compute such prob-
ProbabilisticCognitiveMaps-SemanticsofaCognitiveMapwhentheValuesareAssumedtobeProbabilities
59
ability values with only few values given by experts.
We present briefly three of them.
A
1
α
1
))
. . .
A
i
α
i
. . .
A
n
α
n
uu
B
Figure 6: A concept with many parents.
Some of these methods are dedicated to the repre-
sentation of a cognitive map into a Bayesian network.
(Cheah et al., 2007) provides a procedure that works
only on cognitive maps defined on [−1; 1]. However,
it leads to obtain a probability of 1 in each probabil-
ity table. The combined influence of several parents
may thus be total even if the values of each influence
is low. This problem is obvious when we consider
only two concepts linked by an influence. If the influ-
ence has either a value of 0.1 or 0.9, this value would
be represented by the same value of 1 in the proba-
bility table. Thus, the original influence value is lost.
Note that (Sedki and Bonneau de Beaufort, 2012) uses
a similar method, but with two values on each influ-
ence.
The noisy-OR model (Lemmer and Gossink,
2004) leads to compute the table from individual
conditional probabilities. In this model, the vari-
ables must be binary and the combined influence
of several parents does not matter, as in cognitive
maps. However, it is necessary to suppose that the
given probabilities correspond to the case where
only one parent is set to a specific value and all
the others are set to the opposite value. This
means that we have to give probabilities such as
P(X
B
=+|X
A
1
=−, . . . , X
A
i−1
=−, X
A
i
=+, X
A
i+1
=−,
. . . , X
A
n
=−). This is not consistent with the fact that
the notion of influence is independent from the other
parents.
(Das, 2004) uses a weighted average on many val-
ues. These values and the weights are given by an ex-
pert. Each expert value represents the probability of a
node considering only one of its parents. The weights
represent the relative strengths of the influence of the
parents. This method is suitable for cognitive maps.
The question asked to the expert is indeed: ”Given
that the value of the parent Y is y, compatible with the
values of the other parents, what should be the prob-
ability distribution over the states of the child X?”.
A parent Y
i
with a value y
i
is said compatible with
another parent Y
j
with a value y
j
if, according to the
expert’s mind, the state Y
i
=y
i
is most likely to coexist
with the state Y
j
=y
j
(Das, 2004). This configuration
helps experts to focus only on the state Y
i
=y
i
. We
use this method in our encoding of a cognitive map
as a Bayesian network to fill the probability table of a
node with many parents.
In a cognitive map, the expert values are given by
the influence value. In the previous section, we stated
that the influence value is linked to the difference be-
tween a conditional probability and an a priori prob-
ability. The expert values being considered as condi-
tional probabilities, we define the expert value asso-
ciated to an influence as the sum of the a priori prob-
ability and the converted influence value. With our
example, the expert value of X
B
=+ when X
A
i
=+ is
so 0.5+F (α
i
). Thus, the question to ask to the expert
to get an influence value is: ”Given that A is increas-
ing, this increase being compatible with the states of
the other parents of B, how much the probability that
B is increasing should increase?”. We also stated in
the previous section that the probability of X
B
when
X
A
i
=+ is the complement of the probability of X
B
when X
A
i
=−. Therefore, the expert value of X
B
=+
when X
A
i
=− is 0.5 − F (α
i
).
Besides the values given by the expert, we also
need to provide a weight for each value. However,
in a cognitive map, it is not possible to indicate that
the influence of a concept is more important than the
influence of another one. Thus, the values of the in-
fluences are considered to be evenly important and we
give the same weight for each value.
Definition 7 (Probability table of a concept). Let F
be a conversion function. Let B be a concept and
let X
B
be the random variable associated to B. Let
A
i
∈ C (B) be the parents of B, each one of them be-
ing associated to a random variable X
A
i
. We note, for
each A
i
:
• α
i
= label
(A
i
, B)
;
• a
i
the value of X
A
i
.
The probability table of X
B
is:
P . . . X
A
1
=a
1
, . . . , X
A
n
=a
n
. . .
X
B
=+ 0.5 +
1
n
n
∑
i=1
c(a
i
)
X
B
=− 0.5 −
1
n
n
∑
i=1
c(a
i
)
where c(a
i
) =
F (α
i
) if a
i
= +
−F (α
i
) if a
i
= −
Example 11. Let us consider the node G of CM1 (ex-
ample 1). We give just one example of a computa-
tion of a conditional probability, such as the condi-
tional probability that G is increasing given that S is
decreasing and R is increasing:
P(X
G
=+|X
S
=−, X
R
=+) =
1
2
×
0.5 − F (0.9)
+
1
2
×
0.5 + F (0.8)
= 0.475
The full probability table of the variable X
G
is:
ICAART2014-InternationalConferenceonAgentsandArtificialIntelligence
60
P X
S
=+, X
R
=+ X
S
=+, X
R
=−
X
G
=+ 0.925 0.525
X
G
=− 0.075 0.475
P X
S
=−, X
R
=+ X
S
=−, X
R
=−
X
G
=+ 0.475 0.075
X
G
=− 0.525 0.925
4.2 Relation between a Probabilistic
Propagated Influence and a
Conditional Probability
To ensure that the probabilistic cognitive map model
is valid, we still need to show that our representation
of a cognitive map as a Bayesian network corresponds
to this model. First, we need to clarify what kind
of conditional probability is represented by the prob-
abilistic propagated influence. Let us consider two
concepts A and B. We want to express the link be-
tween the probabilistic propagated influence of A on
B and P(X
B
|X
A
).
Being causal, the reasoning in a cognitive map is
only deductive. The notion of intervention in a causal
Bayesian network leads also to a strictly deductive
reasoning. That’s why this model is closer to the cog-
nitive maps than the classical one: studying the influ-
ence of a concept is indeed similar to intervene on the
value of a variable. With the causal Bayesian network
model, when we are interested in the consequences of
the increase of A on B, the conditional probability is
then P(X
B
=+|do(X
A
=+)).
We stated in the previous section that the notion
of partial probabilistic propagated influence between
A and B is based on the difference between the condi-
tional probability of B given A and the a priori proba-
bility of B. This conditional probability represents in
fact the consequences of an intervention and is thus
P(X
B
=+|do(X
A
=+)). The theorem 1 gives the re-
lation between the partial probabilistic propagated in-
fluence of a concept on another one and the probabil-
ities of the random variables associated to these con-
cepts.
Theorem 1. Let CM be a probabilistic cognitive map.
Let A and B be two concepts of CM. We have:
I
P
0
(A, B) = P(X
B
=+|do(X
A
=+)) − 0.5
Due to a lack of space, we do not give here the
whole proof of this relation. The idea of the proof is
first to define the partial probabilistic propagated in-
fluence as a recursive operator, given by the following
lemma.
Lemma 1. The definition 5 of the partial probabilistic
propagated influence is equivalent to: I
P
0
(c
1
, c
2
) =
0.5 if c
1
= c
2
0 if P
c
1
,c
2
=
/
0
2
|C (c
2
)|
×
∑
c
0
2
∈C (c
2
)
label(c
0
2
, c
2
) × I
P
0
(c
1
, c
0
2
)
otherwise
Then, we prove that in a causal Bayesian
network that represents a cognitive map, any
P(X =+|do(Y =+))−0.5 can also be written as a re-
cursive operator that is trivially equivalent to the one
of lemma 1. We do that by reasoning on a Bayesian
network where the arcs from the parents of Y to Y are
removed and by analysing each possible case, that is
whether X = Y , Y is not a parent of X , Y is a non-
direct parent of X and Y is a direct parent of X. The
full proof is available in a technical report (Le Dorze
et al., 2013).
5 CONCLUSIONS
In this paper, we introduced the new probabilistic cog-
nitive map model where the influence values of a cog-
nitive map are interpreted as probabilities. We defined
consequently the semantics of the concepts and influ-
ences and how to compute the propagated influence
of a concept on another one in such a map. Such a
model gives thus a stronger semantic to the cognitive
maps and provides a better usability for the users. It
also helps to clarify the links between cognitive maps
and Bayesian networks.
Note that the Qualitative Probabilistic Net-
work (QPN) model (Wellman, 1990) is semantically
closer to cognitive maps than Bayesian networks.
M. Wellman considers indeed that the QPNs gener-
alize the cognitive maps. However, the value on each
arc does not quantify a relation between two variables
but simply qualifies it: it expresses constraints be-
tween the probabilities of the many states of the vari-
ables. Some extensions exist to quantify these con-
straints (Renooij and van der Gaag, 2002; Renooij
et al., 2003). Studying if our approach can be related
to QPNs could be interesting.
Last, even if it was not the initial goal, we can see
the work presented in this paper as a first step about
learning Bayesian networks when the information is
expressed by a user with a cognitive map, a cognitive
map being an easy model to capture informal knowl-
edge. Conversely, representing a Bayesian Network
as a cognitive map could help an expert to better un-
derstand the network he has built.
ProbabilisticCognitiveMaps-SemanticsofaCognitiveMapwhentheValuesareAssumedtobeProbabilities
61
REFERENCES
Aguilar, J. (2005). A survey about fuzzy cognitive maps
papers. International Journal of Computational Cog-
nition, 3(2):27–33.
Axelrod, R. M. (1976). Structure of decision: the cognitive
maps of political elites. Princeton, NJ, USA.
Celik, F. D., Ozesmi, U., and Akdogan, A. (2005). Par-
ticipatory Ecosystem Management Planning at Tu-
zla Lake (Turkey) Using Fuzzy Cognitive Mapping.
eprint arXiv:q-bio/0510015.
Charniak, E. and McDermott, D. (1985). Introduction to
Artificial Intelligence. Addison-Wesley, Reading MA.
Chauvin, L., Genest, D., Le Dorze, A., and Loiseau, S.
(2013). User Centered Cognitive Maps. In Guillet, F.,
Pinaud, B., Venturini, G., and Zighed, D. A., editors,
Advances in Knowledge Discovery and Management,
volume 471 of Studies in Computational Intelligence,
pages 203–220. Springer.
Cheah, W. P., Kim, K.-Y., Yang, H.-J., Choi, S.-Y., and Lee,
H.-J. (2007). A manufacturing-environmental model
using Bayesian belief networks for assembly design
decision support. In Okuno, H. G. and Ali, M., edi-
tors, IEA/AIE 2007, volume 4570 of Lecture Notes in
Computer Science, pages 374–383. Springer.
Chickering, D. M. (1996). Learning Bayesian Networks is
NP-Complete. In Learning from Data: Artificial In-
telligence and Statistics V, pages 121–130. Springer-
Verlag.
Das, B. (2004). Generating Conditional Probabilities for
Bayesian Networks: Easing the Knowledge Acquisi-
tion Problem. CoRR, cs.AI/0411034.
Dickerson, J. A. and Bart, K. (1994). Virtual Worlds as
Fuzzy Cognitive Maps. Presence, 3(2):73–89.
Kosko, B. (1986). Fuzzy cognitive maps. International
Journal of Man-Machines Studies, 24:65–75.
Krich
`
ene, J. and Boudriga, N. (2008). Incident Response
Probabilistic Cognitive Maps. In Proceedings of the
IEEE International Symposium on Parallel and Dis-
tributed Processing with Applications (ISPA 2008),
pages 689–694, Los Alamitos, CA, USA. IEEE.
Le Dorze, A., Duval, B., Garcia, L., Genest, D., Leray, P.,
and Loiseau, S. (2013). Probabilistic Cognitive Maps.
Technical report, LERIA – Universit
´
e d’Angers.
Lemmer, J. F. and Gossink, D. E. (2004). Recursive noisy
or - a rule for estimating complex probabilistic inter-
actions. Transactions on Systems, Man, Cybernetics
Part B, 34(6):2252–2261.
Levi, A. and Tetlock, P. E. (1980). A Cognitive Analysis of
Japan’s 1941 Decision for War. The Journal of Con-
flict Resolution, 24(2):195–211.
Nadkarni, S. and Shenoy, P. P. (2001). A Bayesian network
approach to making inferences in causal maps. Euro-
pean Journal of Operational Research, 128(3):479–
498.
Nadkarni, S. and Shenoy, P. P. (2004). A causal mapping
approach to constructing Bayesian networks. Deci-
sion Support Systems, 38(2):259–281.
Pearl, J. (1988). Probabilistic reasoning in intelligent sys-
tems: networks of plausible inference. Morgan Kauf-
mann Publishers Inc., San Francisco, CA, USA.
Pearl, J. (2009). Causality: Models, Reasoning and Infer-
ence. Cambridge University Press, New York, NY,
USA, 2nd edition.
Renooij, S., Parsons, S., and Pardieck, P. (2003). Using
Kappas as Indicators of Strength in Qualitative Proba-
bilistic Networks. In Nielsen, T. D. and Zhang, N. L.,
editors, ECSQARU 2003, volume 2711 of Lecture
Notes in Computer Science, pages 87–99. Springer.
Renooij, S. and van der Gaag, L. C. (2002). From Quali-
tative to Quantitative Probabilistic Networks. In Dar-
wiche, A. and Friedman, N., editors, UAI ’02, pages
422–429. Morgan Kaufmann.
Satur, R. and Liu, Z.-Q. (1999). A Contextual Fuzzy Cogni-
tive Map Framework for Geographic Information Sys-
tems. IEEE Transactions on Fuzzy Systems, 7(5):481–
494.
Sedki, K. and Bonneau de Beaufort, L. (2012). Cognitive
Maps and Bayesian Networks for Knowledge Repre-
sentation and Reasoning. In ICTAI 2012, pages 1035–
1040. IEEE.
Song, H.-J., Shen, Z.-Q., Miao, C.-Y., Liu, Z.-Q., and Miao,
Y. (2006). Probabilistic Fuzzy Cognitive Map. In
FUZZ-IEEE 2006, pages 1221–1228. IEEE.
Spirtes, P., Glymour, C., and Scheines, R. (2001). Causa-
tion, prediction, and search, volume 1. MIT Press.
Tolman, E. C. (1948). Cognitive maps in rats and men. The
Psychological Review, 55(4):189–208.
Wellman, M. P. (1990). Fundamental Concepts of Quali-
tative Probabilistic Networks. Artificial Intelligence,
44(3):257–303.
Zhou, S., Zhang, J. Y., and Liu, Z.-Q. (2003). Quotient
FCMs – A Decomposition Theory for Fuzzy Cog-
nitive Maps. IEEE Transactions on Fuzzy Systems,
11(5):593–604.
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62
