Bayesian Networks in the Presence of Temporal Information - Bayesian Networks in R: With Applications in Systems Biology - page 66

of the sampling time may be critical for meaningful results. It might be interesting,
for example, to infer a network with a time delay from t − 1to t and another one

from t 2to t . If the time delay between two successive time points is too large,

considering a static Bayesian network might be a better choice for the data.
In order to carry out model estimation, it is often assumed that the process is
homogeneous over time (Assumption 4 ). In other words, we assume that the phe-
nomenon we are modeling is governed by the same set of rules during the whole
experiment. Therefore,

−

( − )

repeated measurements are observed for each vari-
able at two successive time points.
Assumption 4 The process is homogeneous over time: all arcs in the network and
their directions are invariant over time.
This allows a good representation of a MTS with a limited number of param-
eters. Each additional time delay would require a specific k

n 1

k coefficient matrix;
therefore, a large number of repeated measurements for each variable at a given time
point would be needed for estimation. However, such data is rarely available. For
instance, most gene expression time series contain no or very few repeated measure-
ments for each gene at a given time point.
While time homogeneity is a strong assumption and not always satisfied for real-
world data, it is often used as a simplifying assumption when the number of obser-
vations is small compared to the number of variables. For completeness, we also dis-
cuss in Sect. 3.4 a recent approach for learning nonhomogeneous dynamic Bayesian
network inference that does not impose homogeneity assumptions.

×

3.2.2 Dynamic Bayesian Network Representation
of a VAR Process

In dynamic Bayesian networks, it is commonly assumed that dependence relation-
ships are represented by a vector auto-regressive process as defined in Eq. 3.5 .A
similar assumption characterized static Gaussian Bayesian networks in Sect. 2.2.4 .
If we assume a VAR process of order 1,

X ( t )= AX ( t − 1 )+ B + ε ( t ) with ε ( t ) ∼ N ( 0 , Σ ) , (3.13)

then all the arcs are defined between two successive time points. The arc set is
defined by the set of nonzero coefficients in A ; if the element a ij , i = j is different

from zero, then the network includes an arc from X i ( .Furthermore,we

assume that the error term for each variable X i is independent from both the other
variables and the respective error terms, so that off-diagonal elements in

t − 1 ) to X j ( t )

can be
set to 0. Of interest is to note that for a VAR(1) process, Assumption 4 is automati-
cally satisfied. The k

Σ

—which has the same nonzero
elements as the adjacency matrix of the interaction network from Fig. 3.2 a—and
the k

× k coefficient matrix A =( a ij )

—representing the baseline measurement for each
variable—are invariant as a function of time. Moreover, Assumption 1 is satisfied
since the random vector X

× 1 column vector B =( b i )

( t ) depends only on the random vector at time ( t − 1 ) .