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Hence, in these approaches the likelihood for cancer in a case is often determined
by the region with the maximum likelihood. However, while the correct detection
and location of a suspicious region is important, in breast cancer screening the
crucial decision based on the mammographic exam is whether or not a patient
should be sent for further examination. Furthermore, most of these approaches
are based on black-box models such as LDA and neural networks, and thus
they lack explanatory power and capability to interpret the classification re-
sults. Therefore, in contrast to previous research, in the current study we aim
at building a probabilistic CAD system, using Bayesian networks, in order to (i)
discriminate well between normal and cancerous cases by considering all avail-
able information (in terms of regions) in a case and (ii) provide an insight in the
automatic mammographic interpretation.
4 Bayesian Networks
4.1 Basic Definitions
Consider a finite set of random variables
X
, where each variable
X
i
in
X
takes
on values from a finite domain
dom
(
X
i
). Let
P
be a joint probability distribution
of
X
and let
S, T, Q
be subsets of
X
.Wesaythat
S
and
T
are conditionally
independent given
Q
, denoted by
S
⊥
P
T
|
Q
, if for all
s
∈
dom
(
S
),
t
∈
dom
(
T
),
q
∈
dom
(
Q
), the following holds:
P
(
s
|
t, q
)=
P
(
s
|
q
), whenever
P
(
t, q
)
>
0
.
In short, we have
P
(
S
Q
).
Bayesian networks, or BNs for short, are used for modelling knowledge in
various domains, from bioinformatics (e.g., gene regulatory networks), to image
processing and decision support systems. A Bayesian network [25] is defined as
apairBN=(
G, P
)where
G
is an acyclic directed graph (ADG) and
P
is a
joint probability distribution. The graph
G
=(
V
,
A
) is represented by a set of
nodes
V
corresponding one to one to the random variables in
X
and a set of arcs
A
⊆
(
V
×
V
) corresponding to direct causal relationships between the variables.
Independence information is modelled in an ADG by blockage of paths between
nodes in the graph by other nodes.
A path between a node
u
and a node
v
in an ADG is
blocked
by a node
w
if the node
w
does
not
have two incoming arcs, i.e.,
|
T,Q
)=
P
(
S
|
,whichis
called a
v-structure
. If there is a v-structure for
w
on the path from
u
to
v
then
the path is
un
blocked by
w
or by one of the descendants of
w
, i.e., nodes that
have a directed path to
w
.Let
U
,
V
,and
W
be sets of nodes, then if any path
between a node in
U
and a node in
V
is blocked by a node in
W
(possibly
the empty set), then
U
and
V
are said to be
d-separated
given
W
, denoted by
U
·−→
w
←− ·
W
. The reader should observe the association of the subscript
G
with
the relation
⊥
G
V
|
. The idea is that the nodes in the sets
U
and
V
are unable to
'exchange' information as any communication path between the nodes in these
sets is blocked. If the sets
U
and
V
are
not
d-separated given
W
,then
U
and
⊥
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