Statistical inference of regulatory networks fro circadian regulation Dirk, H., & Marco, G. (2014). Statistical inference of regulatory networks for circadian regulation. Statistical Applications in Genetics and Molecular Biology. http://dx.doi.org/10.1515/sagmb-2013-0051

1. Statistical inference of regulatory
networks for circadian regulation
— Methodology Part
Zuogong Yue
Pizzaclub, 15th June 2016
Authors: Andrej Aderhold, Dirk Huskier, Marco Grzegorczyk

2. 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

3. 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.

4. Methods
4
o Sparse Regression (LASSO and Elastic Net)
(LASSO)
(Elastic Net)

5. Methods
5
o Time-varying Sparse Regression (Tesla)

6. Methods
6
o Hierarchical Bayesian Regression Model (HBR)
linear regression model:
prior:
then getting the posterior:
and the marginal likelihood:

7. Methods
7
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

8. Methods
8
o Non-homogeneous Hierarchical Bayesian Model
Applying HBR on a multiple change-point process:
Divide the target variable into sub vectors

9. 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

10. Methods
10
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.

11. Methods
11
o Gaussian Processes (GP)
where is the well-known kernel function.
Calculate the posterior:
Marginalize and perform maximization:

12. Methods
12
o Mutual Information Methods (ARACNE)
The mutual information (MI) is given by
A pruning mechanism by Margolin (2006):

13. Methods
13
o Mixture Bayesian Network Models (MBN)
Representing as a Gaussian mixture model (GMM):
Maximize the likelihood of the conditional GMM:

14. Methods
14
o Gaussian Bayesian Network (BGe)
Calculate the posterior distribution of and perform maximization
Assume that
Impose a normal-Wishart prior:

15. 15
Thank you!