Bayesian Networks in the Absence of Temporal Information - Bayesian Networks in R: With Applications in Systems Biology - page 22

Network scores commonly found in literature are the following:

• The Bayesian Dirichlet equivalent (BDe) score, the posterior density associated

with a uniform prior over both the space of the network structures and of the
parameters of each local distribution ( Heckerman et al. , 1995 ).

• The Bayesian information criterion (BIC), a penalized likelihood score defined

as

n
i = 1 log P X i ( X i | Π X i ) −

d
2 log n

BIC = , (2.12)

where d is the number of parameters of the global distribution. It is numerically
equivalent to the information-theoretic minimum description length (MDL) mea-
sure by Rissanen ( 2007 ), even though it has a completely different derivation.
BIC converges asymptotically to the posterior density BDe.

These score functions are said to be score equivalent , since they assign the same
score to networks belonging to the same equivalence class. They are also decom-
posable into the components associated with each node, which is a significant com-
putational advantage when learning the structure of the network (the only parts of
the score that need to be computed are those that differ between the networks being
compared).
In the continuous case, conditional independence tests and network scores are
functions of the partial correlation coefficients ρ XY | Z of X and Y given Z .Two

common conditional independence tests are the following:

• The exact t test for Pearson's correlation coefficient, defined as

)= ρ XY | Z n

− 2

t ( X , Y | Z (2.13)

2
XY

1 − ρ

| Z

and distributed as a Student's t with n −| Z |− 2 degrees of freedom.

• Fisher's Z test, a transformation of the linear correlation coefficient with an

asymptotic normal distribution and defined as

n

+ ρ XY | Z

−| Z |− 3

log 1 Z ( X , Y | Z )=

− ρ XY | Z .

(2.14)

2 1

Both tests can also be performed using Monte Carlo permutation approaches such
as the ones described in Legendre ( 2000 ). Other possible choices are the mu-
tual information test defined in Kullback ( 1968 ), which is proportional to the
corresponding log-likelihood ratio test, or the shrinkage estimators developed by
Shafer and Strimmer ( 2005 ).
Commonly used network scores are again BIC, this time defined as

n
i = 1 log f X i ( X i | Π X i ) −

d
2 log n

BIC = (2.15)