Target Discovery and Validation Reviews and Protocols

Gene Networks 43
where Γ ( . ) is the gamma function. Then, the marginal likelihood p(D|G) can be
analytically computed as
Γ(α j⋅l ) Γ(α jkl + N jkl )
p(D | G) = ∏∏
∏
,
(4)
Γ(α j ⋅l + N j⋅l ) Γ(α jkl )
where α j·l = ∑ α jkl and N By replacing P(D|G) in Eqs. 3 by 4,
k
j ·l = ∑ k Njkl .
we obtain πpost (G|D) for the multinomial distribution and the Dirichlet prior
to perform the structural learning of the Bayesian networks for discrete data.
This criterion is called the BDe metric originally derived by Cooper and
Herskovits (18).
3.2.3. Bayesian Networks for Continuous Data
j
l
Let X j be a continuous type random variable. This situation is more realistic for
gene network inference problem from microarray data, because microarray data
essentially take continuous variables. In this sense, when we apply the Bayesian
network for discrete data to microarray data, we need to discretize expression data
into several categories. The number of categories is usually set to be three, i.e.,
“suppressed,” “unchanged,” and “overexpressed.” However, actually the number
of categories is a parameter to be estimated, and we should notice that the discretization
leads to information loss. Therefore, in microarray data analysis, a computational
method that can handle microarray data as continuous data is advocated.
As for discretization of gene expression data, several methods are listed in Note 6.
The decomposition of the joint probability can be hold for continuous
variables by using densities instead of the probabilistic measure and we have
f (xi1,..., xip |θ) = ∏ f j (xij |pa(X j) i, θ j) ,
j
t t t
θ = (θ1 , ...θp) where f, f1 ,…, fp are densities, is the parameter vector, and
pa(Xj ) i is the set of observations of Pa(Xj ) measured by ith microarray. The
likelihood function is then given by
p(D |θ,G) = ∏ ∏ f j (xij |pa(X j ) i , θ j ) .
i
j
Therefore, in the Bayesian networks for continuous data, the construction of
the conditional densities f j (xij |pa(X j ) i , θ j ) is crucial, and this is essentially
the same as the regression problem.
The linear regression model can be used to construct the conditional density
(6) and is written as
k