Bayesian Networks in the Presence of Temporal Information - Bayesian Networks in R: With Applications in Systems Biology - page 64

acyclicity of the graph, which is required by definition for a Bayesian network. In the
resulting directed acyclic graph, an arc is drawn between two variables at successive
time points, for example, from X 1 ( in Fig. 3.2 c, whenever these two

variables are conditionally dependent given the remaining variables in the past time
points. This condition provides an extension of the properties introduced in Sect. 2.1
for static Bayesian networks and, more in general, of the graphical modeling theory
from Lauritzen ( 1996 ) to temporal data.
In the last decade, various network representations based on different probabilis-
tic models have been proposed in literature: discrete models ( Ong et al. 2002 ; Zou
and Conzen 2005 ), VAR processes ( Opgen-Rhein and Strimmer 2007 ), state-space
or hidden Markov models ( Perrin et al. 2003 ; Wu et al. 2004 ; Rangel et al. 2004 ;
Beal et al. 2005 ), and nonparametric additive regression models ( Imoto et al. 2002 ;
Imoto et al. 2003 ; Kim et al. 2004 ; Sugimoto and Iba 2004 ). We refer the reader to
Kim et al. ( 2003 ) for a comprehensive review of these models. To summarize, we
present in the following a set of sufficient conditions for a model to be represented
as a dynamic Bayesian network; a detailed treatment of these results is provided in
Lebre ( 2009 ).
Consider a dynamic Bayesian network with a directed acyclic graph G (e.g.,
Fig. 3.2 c) which describes a discrete-time stochastic process X

t − 1 ) to X 2 ( t )

= { X i ( t ) ; i = 1 ,..., k ;

k with k variables at n time points. We will show
in Theorem 3.1 that the arc set of this network describes exactly the conditional
dependencies between variables observed at successive time points (e.g., t

t = 1 ,..., t } taking values in R

− 1and t )

given all other variables observed at the earliest time point (e.g., t − 1). This result

rests on the three assumptions below.

Assumption 1 The stochastic process X is first-order Markovian.

Assumption 2 Fo r a l l t > 0 , the random variables X ( t )=( X 1 ( t ) ,..., X i ( t ) ,..., X k

( t )) observed at time t are conditionally independent given the random variables

( − ) −

X t 1 at the previous time t 1 .

Assumption 3 The temporal profile ( X i ( 1 ) ,..., X i ( n )) of any variable X i cannot be

written as a linear combination of the other profiles ( X j ( 1 ) ,..., X j ( n )) , j = i.

Assumption 1 guarantees that any variable at time t is dependent on the past
variables only through the variables observed at time . On the other hand,

Assumption 2 guarantees the variables observed simultaneously at any time point
to be conditionally independent given their immediate past. In other words, time
points are assumed to be close enough that a variable X i at time t is better explained
by X

( t − 1 )

than by other variables X j at the same time t . Therefore, Assumptions 1
and 2 allow the existence of a dynamic Bayesian network with graph G that only
contains arcs pointing out from a variable observed at time

( t − 1 )

toward a variable
observed at time t , with no arcs between simultaneously observed variables. In order
to restrict the number of parameters of the network, we assume a constant time delay
for all interactions, called the time point sampling , defined by the interval between
successive time points. It is certainly possible to add simultaneous interactions, or

( t − 1 )