Biology Reference
In-Depth Information
•
Multinomial variables
: used for discrete/categorical data sets and often referred
to as the
discrete case
. Both the global and the local distributions are multinomial,
and the latter are represented as
conditional probability tables
(CPTs). This is by
far the most common assumption in literature, and the corresponding Bayesian
networks are referred to as
discrete Bayesian networks
.
•
Multivariate normal variables
: this representation is used for continuous data
sets and is therefore referred to as the
continuous case
. The global distribution is
multivariate normal, whereas the local distributions are univariate normal random
variables linked by linear constraints. Local distributions are in fact linear mod-
els in which the parents play the role of explanatory variables. These Bayesian
networks are called
Gaussian Bayesian networks
(
Geiger and Heckerman
,
1994
;
Neapolitan
,
2003
).
Other distributional assumptions require ad hoc learning algorithms or present
various limitations due to the difficulty of specifying the distribution functions in
closed form. For example, models for mixed data, such as the one presented in
Bøttcher and Dethlefsen
(
2003
), impose constraints on the choice of the parents for
the nodes.
On a related note, the choice of a particular set of global and local distributions
also determines which conditional independence tests and which network scores can
be used to learn the structure of the Bayesian network.
Conditional independence tests and network scores for discrete data are functions
of the CPTs implied by the graphical structure of the network through the observed
frequencies
for the random variables
X
and
Y
and all the configurations of the conditioning variables
Z
. Two common
conditional independence tests are the following:
•
{
n
ijk
,
i
=
1
,...,
R
,
j
=
1
,...,
C
,
k
=
1
,...,
L
}
Mutual information
(
Cover and Thomas
,
2006
), an information-theoretic dis-
tance measure defined as
R
i
=
1
C
j
=
1
L
k
=
1
n
ijk
n
log
n
ijk
n
++
k
MI
(
X
,
Y
|
Z
)=
n
i
+
k
n
+
jk
.
(2.10)
It is proportional to the log-likelihood ratio test G
2
(they differ by a 2
n
factor,
where
n
is the sample size), and it is related to the deviance of the tested models.
The classic
Pearson's X
2
test for contingency tables,
•
n
ijk
−
m
ijk
2
m
ijk
R
i
=
1
C
j
=
1
L
k
=
1
n
i
+
k
n
+
jk
n
++
k
X
2
(
,
|
)=
,
m
ijk
=
.
X
Y
Z
where
(2.11)
In both cases the null hypothesis of independence can be tested using either the
asymptotic
2
(
L
distribution or the Monte Carlo permutation approach
described in
Edwards
(
2000
). Other possible choices are Fisher's exact test and the
shrinkage estimator for the mutual information defined by
Hausser and Strimmer
(
2009
) and studied in
Scutari and Brogini
(
2012
).
χ
R
−
1
)(
C
−
1
)
Search WWH ::
Custom Search