Target Discovery and Validation Reviews and Protocols

Gene Networks 41
Fig. 6. Directed acyclic graph.
In gene network inference by Bayesian networks, a gene is regarded as a
random variable and represented by a node in a dag, and the relationship
between a gene and its direct parents is represented as a conditional probability
Pr(X j |Pa(X j )). Boolean networks are a deterministic system that explains a
state of a gene by the Boolean function of the states of its direct parents. On
the contrary, the Bayesian networks use a probabilistic nature to model the
dependencies among genes. Because gene expression data measured by
microarrays contain various kinds of noise, the probabilistic formulation in the
Bayesian networks is advocated to draw more reliable information from
microarray data.
The decomposition in Eq. 1 holds when the structure of the directed acyclic
graph G is given. However, for gene network estimation problem from microarray
data, many parts of the true structure of the gene network are still unknown, and
we need to estimate them from microarray data. This structural learning problem
can be viewed as a statistical model selection problem (17) and can be
solved by using an information criterion, such as the Akaike information criterion,
the Bayesian information criterion, and so on. In this chapter, we describe
a method to select a network structure based on the Bayes approach. In the
Bayes statistics, the optimal graph can be selected as the maximizer of the
posterior probability of the graph conditional on the data.
Suppose we have n independent observations D = {x 1 ,…,x n } for {X 1 ,…,X p },
where the ith observation x i = (x i1 ,…,x ip ) t is the p-dimensional vector. Here a t
indicates the transpose of the vector a. In our case, the posterior probability of
the graph can be written as follows:
π post(G | D) = π prior (G)p(D | G)
,
p(D)
where πprior (G) is the prior probability of the graph, p(D) is the normalizing
constant given by ∑ π (G)p(D | G) and p(D | G) is the marginal likelihood
G prior
obtained by