Target Discovery and Validation Reviews and Protocols

44 Imoto et al.
β k (k = 1,…,q j ) are parameters and ε ij (i = 1,…,n) are independently and normally
distributed with mean 0 and variance σ j 2 . This model assumes the relationships
between genes are linear, and it is unsuitable to extract effective
information from the data with complex structure. To capture even nonlinear
dependencies, Imoto et al. (9) proposed the use of the nonparametric additive
regression model (19) of the form
where m jk ( . ) (k =1,…,q j ) are smooth functions from R to R. We construct m jk ( . )
by the basis function expansion method with B-splines (20,21):
where are parameters, is the prescribed
set of B-splines, and Mjk is the number of B-splines. We then have the
Bayesian network and nonparametric regression model
f j (xij | pa(X j ) i ,θ j ) =
( j ) ( j ) (j )
1 x − γ
ij ∑ bsk ( pik
{ ∑
)
k s sk } exp −
2
2πσ j
2
( j )
γ sk (s = 1,…,M jk )
⎡
⎤
⎢
⎥
⎢
2
⎥.
⎢
2σ j
⎥
⎣
⎦
Note that the Bayesian network model based on the linear regression is
included in this model as a special case. To extend from the additive regression
model to a general regression, see Note 7.
Let the prior distribution on the parameter be specified by the hyperparameter
and let log p( | ) = O(n), then the Laplace approximation for integrals
(22–24) gives an analytical solution for computing p(D|G) as follows:
where
( j) ( j ) ( j) ( j) t
xij =β0 +β1p +⋅⋅⋅+βqj i1 piq +εij , where pa(X
j
j ) i = (pi1 ,…, piqj ) ,
( j ) ( j )
xij = m j1 (pi 1 ) +⋅⋅⋅+m (p jqj iq )+εij ,
j
p(D |G) = (2π / n) r /2 | J λ ( ˆ θ | D)| –1/2 exp{nl λ( ˆ
θ | D)}{1+O p (n −1 )},
l λ (θ | D) = 1
n
M jk
∑
m jk (p) = γ sk
s =1
{log p(D | θ, G)+ log p(θ | λ)},
J λ (θ | D) = –
( j ) ( j )
bsk (p),
∂ 2
∂θ∂θ t l λ
( j) ( j)
{b1k (⋅),⋅⋅⋅⋅⋅bM
jk k (⋅)}
(θ |D),