Statistical inference of regulatory networks for circadian regulation

— Methodology Part

Zuogong Yue

Pizzaclub, 15th June 2016

Authors: Andrej Aderhold, Dirk Huskier, Marco Grzegorczyk

o Mathematical formulation of transcriptional regulation1

Problem Formulation

2

1 Barenco, M., Tomescu, D., Brewer, D., Callard, R., Stark, J., & Hubank, M. (2006). Ranked prediction of p53 targets using hidden variable dynamic modeling. Genome Biology, 7(3), R25.

o Regulatory networks (bipartite structure)

yg1

yg2

yg3

xg1

xg2

xg3

xg4

xg5

Methods

3

o Graphical Gaussian Models (GGM)

The components corresponding to two genes are stochastically independent conditional on the remaining system

if and only if the corresponding element in the inverse covariance matrix is zero.

Methods

4

o Sparse Regression (LASSO and Elastic Net)

(LASSO)

(Elastic Net)

Methods

5

o Time-varying Sparse Regression (Tesla)

Methods

6

o Hierarchical Bayesian Regression Model (HBR)

linear regression model:

prior:

then getting the posterior:

and the marginal likelihood:

Methods

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o Hierarchical Bayesian Regression Model (HBR) (cont.)

Finally we get the marginal posterior distribution on

Maximizing the above posterior by Markov chain Monte Carlo (MCMC)

and

Methods

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o Non-homogeneous Hierarchical Bayesian Model

Applying HBR on a multiple change-point process:

Divide the target variable into sub vectors

Methods

9

o Automatic Relevance Determination (ARD) - Sparse Bayesian Regression (SBR)

Using the prior distribution: (choosing appropriate hyper parameters can lead to sparse solutions):

The marginal likelihood:

Maximize the marginal likelihood by Expectation Maximization (EM) method

Methods

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o Bayesian Spline Autoregression (BSA)

The original covariates are augmented with B-spline basis functions.

To encourage network sparsity, a slab-and-stick-like Bayesian variable selection scheme2 is used.

2 Smith, M., & Kohn, R. (1996). Nonparametric regression using Bayesian variable selection. Journal of Econometrics, 75(2), 317–343.

Methods

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o Gaussian Processes (GP)

where is the well-known kernel function.

Calculate the posterior:

Marginalize and perform maximization:

Methods

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o Mutual Information Methods (ARACNE)

The mutual information (MI) is given by

A pruning mechanism by Margolin (2006):

Methods

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o Mixture Bayesian Network Models (MBN)

Representing as a Gaussian mixture model (GMM):

Maximize the likelihood of the conditional GMM:

Methods

14

o Gaussian Bayesian Network (BGe)

Calculate the posterior distribution of and perform maximization

Assume that

Impose a normal-Wishart prior:

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Thank you!