Target Discovery and Validation Reviews and Protocols

42 Imoto et al.
p(D |G) = ∫ p(D,θ | G)dθ = ∫ p(D |θ,G)p(θ |G)dθ .
Here, p(D | , G) is the likelihood function, and p( | G) is the prior distribution
on the parameter . Because the normalizing constant p(D) does not depend on
G, we can find
π post (G|D) ∝ π prior (G)p(D|G).
Hence, for computing the posterior probability for structural learning of the
Bayesian networks, we need to compute the high-dimensional integral in Eq. 2.
For learning Bayesian networks, Note 5 gives some information.
3.2.2. Bayesian Networks for Discrete Data
Suppose that we have a set of random variables {X 1 ,…,X p } and that a random
variable takes one of m values {u 1 ,…,u m }. Then, we put the conditional
probability as
θ jkl = Pr (X j = u k | Pa(X j ) = u jl )
for j = 1,…,p; k = 1,…,m; l = 1,…,m |Pa (X j )| m
. Note that ∑ = 1 holds.
Here, ujl is the lth entry of the state table of Pa(Xj ). For example, in Fig. 4,
Pa(X1 ) = {X2 , X3 } contains m2 θ k =1 jkl
entries: u11 = (u1 , u1 ), u12 = (u1 , u2 ),…,u1m2 =
(um , um ). In this case, we can assume that Xj |P a(Xj ) = ujl follows the multinomial
distribution with probabilities θj1l ,…,θjml .
Suppose we have n independent observations D = {x1 ,…, xn } for {X1 ,…,Xp },
where xij is one of the m values {u1 ,…, um }. Based on these observations, the
likelihood function of θ = (θjkl ) j,k,l is written by
N jkl
p(D |θ, G) = ∏ ∏∏θ
jkl ,
where Njkl is the number of pairs satisfying (Xj , Pa(Xj )) = (uk , ujl ) in D. In the
computation of the marginal likelihood p(D|G) given in Eq. 2 for the Bayesian
networks for discrete data, the Dirichlet distribution is usually used as the prior
distribution on the parameter p(|G). Suppose that the prior distribution on can
be decomposed as where jl =(θj1l ,…,θjml ) t
and jl = (αj1l ,…,αjml ) t p(θ | G) = ∏ j ∏ l p(θ jl | α jl ),
. Here, jl is called a hyperparameter vector. We assume
the Dirichlet prior for p(jl |jl ) as
p(θ jl|α jl) = Γ(α jkl )
Γ( kl )
∏ αj ′
k'
j
k
∏ θ
k
l
α jkl −1
jkl
,
(2)
(3)