Using Bayesian Networks to
Analyze Expression Data
Jeong, Jong Cheol
Nir Friedman, Michal Linial, Iftach Nachman & Dana Pe’er
Dept. Electrical Engineering & Computer Science
University of Kansas
Bayesian Networks
Representation of a joint probability
distribution
A directed acyclic graph:

Random variables

Conditional distribution
Conditional independence assumption

Each variable is independent of its none

descendants
Conditional independence assumption
Any joint distribution can be decomposed
into product form
1
1
(,...,) (  ( ))
n
G
n i i
i
P X X P X X
Pa
( )
G
i
X
Pa
i
X
is the set of parents of
Bayesian Networks
Conditional independence
I(A; E), I(B; D  A, E),
I(C; A, D, E  B)
I(D; B, C, E  A), I(E; A, D)
P(A,B,C,D,E) = P(A)∙P(BA,E) ∙P(CB) ∙P(DA) ∙P(E)
Specifying conditional distribution
Discrete variables:

Using table that specifies the probability of values

For binary variable, the table specifies
distribution
Continuous variables:

Using linear Gaussian distribution
1
( ,...)
k
P X U U
2
k
1
{,...}
k
U U
: parents of a variable of
2
( ,...,) (,)
i k o i i
i
P X u u N a a u
Equivalence Classes of Bayesian Networks
Two directed acyclic graphs are equivalent if only if
they have the same underlying undirected graph and
the same v

structure (Pearl & Verma 1991).

converging directed edges into the same node:
a b c
Ind(G): the set of independence statements

if more than one graph exactly same sat of
independencies
where
How can be distinguish between equivalent graph?
:
G X Y
':
G X Y
Ind( ) Ind(')
G G
Learning Bayesian Networks
Training set:

finding a network
which best matches D
Score function

evaluating the posterior probability of a graph given
the data
1
{,...,}
N
D
x x
,,
B G
θ
(  ( ))
G
i i
P X X
Pa
(:) log (  )
log (  ) log ( )
S G D P G D
P D G P G C
(  ) ( ,) (  )
P D G P D G P G d

Marginal likelihood
Property of priors
Structure equivalent:
if graph G and G’ are
equivalent graphs then they are guaranteed to have
the same posterior score.
Decomposable:
the contribution of variable to
the total network score depends only on its own value
and the values of its parents in G
Local contributions for each variable can be
computed using a closed form equation.
i
X
(:) ScoreContribution(,( ):)
G
i i
i
S G D X X D
Pa
Learning Causal Patterns
Bayesian network
:
model of dependencies
between multiple measurements
.
A causal network
:
having stricter interpretation of
the meaning of edges (i.e., the parents of a variable are
its immediate causes.)
X Y X Y
Using Bayesian Networks to Analyze
Expression Data
(Nir Friedman et. al.)
Goal

Building Bayesian networks which can be applied to
model interactions among genes

Examine
1. Markov relation
2. Order relations
Estimating Statistical Confidence in
Features
Using bootstrap method
for
1...
i m
i
D
: sampling N instances from D with replacement
Apply the learning procedure on to induce a network structure
i
D
i
G
f
for each feature , calculating
1
1
conf ( ) ( )
m
i
i
f f G
m
where
1 : is a feature in G
( )
0 : otherwise
i
f
f G
Sparse Candidate algorithm
Choosing most promising candidate parents for
i
X
1
{,...,}
n
i k
C Y Y
Searching a high network in which
( )
n
G
n
i i
X C
Pa
Repeat
if
2 1
( ) ( )
n n
Score G Score G
1
n n
G G
End if
Until
is no changeable
n
i
C
Measuring the relevance of potential parent
to .
j
X
i
X
1
1
ScoreContribution(,( ) { }:)
ScoreContribution(,( ):)
n
n
G
i i j
G
i i
X Pa X X D
X Pa X D
Application to Cell Cycle Expression
Patterns
Data set: S. cerevisiae ORFs

76 gene expression measurements of the mRNA
levels of 6177.
Sparse candidate algorithm with 200

fold bootstrap
Experiment

the discrete multinomial distribution

linear Gaussian distribution
Markov features
Local map for the gene SVS1
Multinomial
Linear Gaussian
References
Friedman, N., Linial, M., Nachman, I., & Pe’er, D. Using bayesian
networks to analyze expression data.
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