1. Dynamic Bayesian Networks
Sushmita Roy
Computational Network Biology Biostatistics & Medical Informatics 826 Computer Sciences 838
Oct 18th 2016

2. Goals for today Dynamic Bayesian Network
How does a DBN differ from a static BN?
Learning a Dynamic Bayesian Network
DBN with structure priors
Application of DBN to a phospho-proteomics time course
Evaluation and insights

3. Dynamic Bayesian Networks
A Dynamic Bayesian network (DBN) is a Bayesian network that can model temporal/sequential data
DBN is a Bayes net for dynamic processes
A DBN also has a graph structure and conditional probability distributions
The DBN specifies how observations at a future time point may arise from previous time points.

4. Notation
Assume we have a time course with T time points specifying activity of p different variables
Let denote the set of random variables at time t
A DBN over these variables defines the joint distribution of P(X), where
A DBN, like a BN, has a directed acyclic graph G and parameters Θ
G typically specifies the dependencies between time points
In addition we need to specify dependence (if any) at t=0

5. A DBN for p variables and T time points
X11
X21
Xp1 … X1 X2 Xp … 1 X1 X2 Xp … 2 … X1 X2 Xp … T … p …
X2: Variables at time t=2
Dependency at the first time point

6. Stationary assumption in a Bayesian network
The stationarity assumption states that the dependency structure and parameters do not change with t
Due to this assumption, we only need to specify dependencies between two sets of variables p t
X1t
X2t
Xpt …
X1t+1
X2t+1
Xpt+1
t+1 X1 X2 Xp 1 2 T
t=1
t=2
t=T

7. Computing the joint probability distribution in a DBN
Joint Probability Distribution can be factored into a product of conditional distributions across time and variables:
Parameters specifying the form of the conditional distributions
Graph encoding dependency structure between variables at consecutive time points
Parents of Xit defined by the graph G

8. Learning problems in DBNs
Parameter learning: Given known temporal dependencies between random variables estimate the parameters from observed measurements
Structure learning: Given data, learn both the graph structure and parameters
Complexity of learning depends upon the order of the model

9. An example DBN
Let us consider a simple example of two regulators, B and C and one target gene, A
Assume their expression takes on values H, L and NC (for high, low and no-change in expression)
A’s expression level depends upon regulator B and C’s expression level
B and C mutually regulate each other
Let XAt denote the random variable representing the expression level of gene A at time t

10. DBN for a three node network
The collapsed network

11. Specifying the parameters of the DBN for a three node network
Each of these conditional distributions will specify the distribution over {H,L, NC} given the state of the parent variable
DBN

12. Specifying the parameters of the DBNL NC
0.5
0.1
0.4
0.2
0.25

13. Specifying the parameters of the DBNL NC
0.8
0.1
0.2
0.6
0.7
0.3
0.05
0.75
Parameter estimation: Estimating these numbers

14. Assume the following CPDs for three variablesNC
0.5
0.1
0.4
0.2
0.25 B C H L NC
0.8
0.1
0.2
0.6
0.7
0.3
0.05
0.75 H L NC
0.5
0.1
0.4
0.2
0.25

15. Computing the probability distribution of an observation
Suppose we are given a new observation time course
T=0
T=1
T=2 NC L H
Assume, P(NC)=0.5 and P(H)=P(L)=0.25 for all variables at T=0.
Using the DBN from the previous slides, what is the probability of this time course?
First we plug in the formula at the time point level
Next, we look at the graph structure of the DBN to further decompose these terms

16. Computing the probability distribution of an observation
Graph structure of the DBN to further decompose these terms NC L H
Assume P(NC)=0.5 and P(H)=P(L)=0.25 at T=0.

17. Parameter estimation in DBNs
Parameter estimation approach would differ depending upon the form of the CPD
Assume that the variables are discrete, then we need to estimate the entries of the CPD distribution

18. Parameter estimation example for three node DBN
Need to estimate this table H L NC
Suppose we had a training time course:
T=0
T=1
T=2
T=3
T=4 NC L H
To compute these probabilities, we need to look at the joint assignments of {XBt+1,XCt} for all 0≤t≤4
What is P(XBt+1=H|XCt=L)?
What is P(XBt+1=NC|XCt=L)?

19. Structure learning in DBNs
We need to learn the dependency structure between two consecutive time points
We may also want to learn within time point connectivity
Structure search learning algorithms used for BNs, can be used with a simple extension:
parents of a node can come from the previous or current time step.

20. DBN with score-based search
Score of a DBN is a function of the data likelihood
Data: Collection of time courses
Graph prior: This can be uniform,
or can encode some form of model complexity

21. Goals for today Dynamic Bayesian Network
How does a DBN differ from a static BN?
Learning a Dynamic Bayesian Network
DBN with structure priors
Application of DBN to a phospho-proteomics time course
Evaluation and insights

22. Bayesian Inference of Signaling Network Topology in a Cancer Cell Line (Hill et al 2012)
Protein signaling networks are important for many cellular diseases
The networks can differ between normal and disease cell types
But the structure of the network remains incomplete
Temporal activity of interesting proteins can be measured over time, that can be used infer the network structure
Build on prior knowledge of signaling networks to learn a better, predictive network
BNs are limiting because they do not model time

23. Applying DBNs to infer signaling network topology
Fig.1: Data-driven characterization of signaling networks. Reverse-phase protein arrays interrogate signaling dynamics in samples of interest. Network structure is inferred using DBNs, with primary phospho-proteomic data integrated with existing biology, using informative priors objectively weighted by an empirical Bayes approach. Edge probabilities then allow the generation and prioritization of hypotheses for experimental validation
Hill et al., Bioinformatics 2012

24. Application of DBNs to signaling networks
Dataset description
Phospho-protein levels of 20 proteins
Eight time points
Four growth conditions
Use known signaling network as a graph prior
Estimate CPDs as conditional regularized Gaussians
Assume a first-order Markov model
Xt depends on on Xt-1

25. Integrating prior signaling network into the DBN
A Bayesian approach to graph learning
Graph prior is encoded as (Following Mukherjee & Speed 2008)
Where f(G)=-|E(G)\E*| is defined as the number of edges in the graph G, E(G), that are not in the prior set E*
This prior does not promote new edges, but penalizes edges that are not in the prior
Data likelihood
Graph prior
Prior strength
Graph features

26. Calculating posterior probabilities of edges
For each edge e, we need to calculate
Although this is intractable in general, this work makes some assumptions
Allow edges only forward in time
The learning problem decomposes to smaller per-variable problems that can be solved by variable selection
Assume P(G) factorizes over individual edges
To compute the posterior probability, the sum goes over all possible parent sets
Assume a node can have no more than dmax parents

27. Results on simulated data
20 variables, 4 time-courses
8 time points
Prior network had 54 extra edges and did not have 10 of the ground truth edges

28. Results are not sensitive to prior values
Sensitivity to choice of hyper parameter
Sensitivity to noisy prior graph

29. Inferred signaling network using a DBN
Prior also had self-loops that are not shown
Inferred signaling network
Prior network

30. Using the DBN to make predictions
Although many edges were expected, several edges were unexpected
Select novel edges based on posterior probability and test them based on inhibitors
For example, if an edge was observed from X to Y, inhibition of X should affect the value of Y, if X is a regulator of Y
Example edges tested
MAPKp to STAT3p(S727) with high probability (0.98)
Apply MEKi, which is an inhibitor of MAPK, and measure MAPKp and STAT3p post inhibition
AKTp to p70S6Kp, AKTp to MEKp and AKTp to cJUNp

31. Experimental validation of links
Add MAPK inhibitor and measure MAPK and STAT3
MAPK is significantly inhibited (P-value 5X10-4)
STAT3 is also inhibited (P-value 3.3X10-4)
Fig. 4. Validation of predictions by targeted inhibition in breast cancer cell line MDA-MB-468. (a) MAPK-STAT3 crosstalk. Network inference (Fig. 3a) predicted an unexpected link between phospho-MAPK (MAPKp) and STAT3p(S727) in the breast cancer cell line MDA-MB-468. The hypothesis of MAPK-STAT3 crosstalk was tested by MEK inhibition: this successfully reduced MAPK phosphorylation and resulted in a corresponding decrease in STAT3p(S727). (b) AKTp → p70S6Kp, AKT-MAPK crosstalk and AKT-JNK/JUN crosstalk. AKTp is linked to p70S6kp, MEKp and cJUNp. In line with these model predictions, use of an AKT inhibitor reduced both p70S6K and MEK phosphorylation and increased JNK phosphorylation. (RPPA data; MEK inhibitor GSK and AKT inhibitor GSK690693B at 0 uM, uM, 2.5 uMand 10 uM; measurements taken 0, 5, 15, 30, 60, 90, 120 and 180min after EGF stimulation; average values over 3 replicates shown, error bars indicate SEM)
Their success is measured by the difference in the levels of the targets as a function of the levels of the inhibitors

32. Take away points Network dynamics can be defined in multiple ways
We have seen two ways to capture network dynamics
Skeleton network-based approaches
The universe of networks is fixed
Nodes become on or off
No assumption or model of how the network changes over time
Dynamic Bayesian network
A type of probabilistic graphical model
Describes how the system transitions from one state to another
Assumes that the dependency between t-1 and t is the same for all time points
Application to cancer signaling data
DBNs are powerful for capturing the dynamics
However, the prior was important to learn an accurate network

33. References
N. Friedman, K. Murphy, and S. Russell, "Learning the structure of dynamic probabilistic networks," in Proceedings of the Fourteenth Conference on Uncertainty in Artificial Intelligence, ser. UAI'98.    San Francisco, CA, USA: Morgan Kaufmann Publishers Inc., 1998, pp [Online]. Available:
S. M. Hill, Y. Lu, J. Molina, L. M. Heiser, P. T. Spellman, T. P. Speed, J. W. Gray, G. B. Mills, and S. Mukherjee, "Bayesian inference of signaling network topology in a cancer cell line." Bioinformatics (Oxford, England), vol. 28, no. 21, pp , Nov [Online]. Available: