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Evaluation

Problem

- Limited number of experimental replications.
- Postgenomic data intrinsically noisy.
- Poor network reconstruction.

Problem

- Limited number of experimental replications.
- Postgenomic data intrinsically noisy.
- Can we improve the network reconstruction by systematically integrating different sources of biological prior knowledge?

+

Which sources of prior knowledge are reliable?

- How do we trade off the different sources of prior knowledge against each other and against the data?

Overview of the talk

- Revision: Bayesian networks
- Integration of prior knowledge
- Empirical evaluation

Overview of the talk

- Revision: Bayesian networks
- Integration of prior knowledge
- Empirical evaluation

- Marriage between graph theory and probability theory.
- Directed acyclic graph (DAG) representing conditional independence relations.
- It is possible to score a network in light of the data: P(D|M), D:data, M: network structure.
- We can infer how well a particular network explains the observed data.

NODES

A

B

C

EDGES

D

E

F

Bayesian networks versus causal networks

Bayesian networks represent conditional (in)dependence relations - not necessarily causal interactions.

Bayesian networks versus causal networks

A

A

A

B

C

B

C

B

C

- Equivalence classes: networks with the same scores: P(D|M).
- Equivalent networks cannot be distinguished in light of the data.

A

B

C

Prior knowledge:

B is a transcription factor with binding sites in the upstream regions of A and C

Overview of the talk

- Revision: Bayesian networks
- Integration of prior knowledge
- Empirical evaluation

Biological prior knowledge matrix

Indicates some knowledge about

the relationship between genes i and j

Biological Prior Knowledge

Biological prior knowledge matrix

Indicates some knowledge about

the relationship between genes i and j

Biological Prior Knowledge

Define the energy of a Graph G

Notation

- Prior knowledge matrix:

P B (for “belief”)

- Network structure:

G (for “graph”) or M (for “model”)

- P: Probabilities

Prior distribution over networks

Sample networks and hyperparameters

- from the posterior distribution
- Capture intrinsic inference uncertainty
- Learn the trade-off parameters automatically

P(M|D) = P(D|M) P(M) / Z

Prior distribution over networks

Rewriting the energy

Approximation of the partition function

Partition functionof a perfect gas

Sample networks and hyperparameters from the posterior distribution

Proposal probabilities

Metropolis-Hastings scheme

Bayesian networkswith biological prior knowledge

- Biological prior knowledge: Information about the interactions between the nodes.
- We use two distinct sources of biological prior knowledge.
- Each source of biological prior knowledge is associated with its own trade-off parameter:b1 and b2.
- The trade off parameter indicates how much biological prior information is used.
- The trade-off parameters are inferred. They are not set by the user!

Bayesian networkswith two sources of prior

Source 2

Source 1

Data

BNs + MCMC

b1

b2

Recovered Networks and trade off parameters

Bayesian networkswith two sources of prior

Source 2

Source 1

Data

BNs + MCMC

b1

b2

Recovered Networks and trade off parameters

Bayesian networkswith two sources of prior

Source 2

Source 1

Data

BNs + MCMC

b1

b2

Recovered Networks and trade off parameters

Overview of the talk

- Revision: Bayesian networks
- Integration of prior knowledge
- Empirical evaluation

Evaluation

- Can the method automatically evaluate how useful the different sources of prior knowledge are?
- Do we get an improvement in the regulatory network reconstruction?
- Is this improvement optimal?

From Sachs et al Science 2005

Evaluation: Raf signalling pathway

- Cellular signalling network of 11 phosphorylated proteins and phospholipids in human immune systems cell
- Deregulation carcinogenesis
- Extensively studied in the literature gold standard network

- Intracellular multicolour flow cytometry experiments: concentrations of 11 proteins
- 5400 cells have been measured under 9 different cellular conditions (cues)
- Downsampling to 100 instances (5 separate subsets): indicative of microarray experiments

Spellman et al (1998)

Cell cycle

73 samples

Tu et al (2005)

Metabolic cycle

36 samples

time

time

Genes

Genes

http://www.genome.jp/kegg/

KEGG PATHWAYS are a collection of manually drawn pathway maps representing our knowledge of molecular interactions and reaction networks.

Evaluation

- Can the method automatically evaluate how useful the different sources of prior knowledge are?
- Do we get an improvement in the regulatory network reconstruction?
- Is this improvement optimal?

Bayesian networkswith two sources of prior

Source 2

Source 1

Data

BNs + MCMC

b1

b2

Recovered Networks and trade off parameters

Bayesian networkswith two sources of prior

Source 2

Source 1

Data

BNs + MCMC

b1

b2

Recovered Networks and trade off parameters

Evaluation

- Can the method automatically evaluate how useful the different sources of prior knowledge are?
- Do we get an improvement in the regulatory network reconstruction?
- Is this improvement optimal?

- Can the method automatically evaluate how useful the different sources of prior knowledge are?
- Do we get an improvement in the regulatory network reconstruction?
- Is this improvement optimal?

Learning the trade-off hyperparameter

Mean and standard deviation of the sampled trade off parameter

- Repeat MCMC simulations for large set of fixed hyperparameters β
- Obtain AUC scores for each value of β
- Compare with the proposed scheme in which β is automatically inferred.

Conclusion

- Bayesian scheme for the systematic integration of different sources of biological prior knowledge.
- The method can automatically evaluate how useful the different sources of prior knowledge are.
- We get an improvement in the regulatory network reconstruction.
- This improvement is close to optimal.

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