Biology Reference
In-Depth Information
b
a
X
1
X
2
X
3
X
1
X
2
X
3
X
1
X
2
X
3
c
X
1
(2)
X
1
(t)
X
1
(n)
X
1
(1)
X
2
(1)
X
2
(2)
X
2
(t)
X
2
(n)
X
3
(1)
X
3
(2)
X
3
(t)
X
3
(n)
Fig. 3.2
Graphical representation of a static Bayesian network, a dynamic Bayesian network, and
a time-varying dynamic Bayesian network.
(a)
An example of an interaction network between
three variables
X
1
,
X
2
,
X
3
modeling a genetic regulatory motif between genes. Arcs ending with
an arrow correspond to gene activations, and arcs ending with a line correspond to gene inhibi-
tions.
(b)
Because Bayesian networks are constrained to be acyclic, they cannot contain loops
or cycles. Therefore, the motif in (a) cannot be correctly modeled by a conventional (static)
Bayesian network which does not take temporal ordering into account. If
X
3
is conditionally
independent from
X
1
given
X
2
, we have that both P
(
X
1
,
X
2
,
X
3
)=
P
(
X
3
|
X
2
)
P
(
X
2
|
X
1
)
P
(
X
1
)
and
P
are valid sets of local distributions.
(c)
If time-course
expression measurements are available, it is possible to unravel the feedback cycles and the loops
over time points. Such time-homogeneous dynamic Bayesian network assumes that at each given
time
t
, all the parents of each node are measured at the previous time point
t
−
1
(
X
1
,
X
2
,
X
3
)=
P
(
X
3
|
X
2
)
P
(
X
1
|
X
2
)
P
(
X
2
)
a longer time delay, by allowing the existence of arcs between variables observed
either at the same time
t
or with a longer time delay (i.e., from
t
−
2to
t
). However,
the number of parameters of the model increases exponentially with the number
of time delays, which can be challenging given the number of time points in most
data sets.
Finally, Assumption
3
guarantees the uniqueness of
G
when the
k
variables are
linearly independent, i.e., none of the profiles can be written as a linear combination
of the others. When these three assumptions are satisfied, the probability distribution
of the process
X
can be represented by a dynamic Bayesian network as described
by the following theorem.
Theorem 3.1.
Whenever Assumptions
1
,
2
, and
3
are satisfied, the probability dis-
tribution of
X
can be represented as a dynamic Bayesian network with a directed
acyclic graph G whose arcs describe exactly the conditional dependencies between
any pair of variables
(
X
j
(
t
−
1
)
,
X
i
(
t
))
at successive time points given the past vari-
ables X
(
t
−
1
)
\{
X
j
(
t
−
1
)
}
.
As expected, dynamic Bayesian network models are dependent on the sampling
time and the choice of the time delay. Interactions that occur at a time scale shorter
than the sampling time may not necessarily be detected from the given data and
can lead to spurious conclusions on the network structure. Thus, a prudent choice
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