Using Bayesian Networks to Analyze Expression Data

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Using Bayesian Networks to Analyze Expression Data
Nir Friedman, Michal Linial, Iftach Nachman
Dana Peer

• Jeong, Jong Cheol

Dept. Electrical Engineering Computer
Science University of Kansas
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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

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Conditional independence assumption

• Any joint distribution can be decomposed into
  product form

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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)
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Specifying conditional distribution
parents of a variable of

• Discrete variables- Using table that specifies
  the probability of values
• - For binary variable, the table specifies
  distribution
• Continuous variables- Using linear Gaussian
  distribution

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Equivalence Classes of Bayesian Networks

• Ind(G) the set of independence statements- if
  more than one graph exactly same sat of
  independencieswhereHow can be distinguish
  between equivalent graph?

• 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

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Learning Bayesian Networks

• Training set - finding a network which
  best matches D
• Score function- evaluating the posterior
  probability of a graph given the data

- Marginal likelihood
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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.

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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.)

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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

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Estimating Statistical Confidence in Features

• Using bootstrap method

for
sampling N instances from D with replacement
Apply the learning procedure on to induce a
network structure
for each feature , calculating
where
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Sparse Candidate algorithm
Repeat
Choosing most promising candidate parents for
Searching a high network in which
if
End if
Until is no changeable
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• Measuring the relevance of potential parent
  to .

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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

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Markov features
Local map for the gene SVS1
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Multinomial
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Linear Gaussian
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References

• Friedman, N., Linial, M., Nachman, I., Peer,
  D. Using bayesian networks to analyze expression
  data.