1. Temporal Models Template Models Representation Probabilistic Graphical

2. Distributions over Trajectories1 2 3 4 5
Pick time granularity 
X(t) – variable X at time t
X(t:t’) = {X(t), …, X(t’)} (t  t’)
Want to represent P(X(t:t’)) for any t, t’

3. Markov Assumption
Counter example: location and velocity

4. Time Invariance Template probability model P(X’ | X) For all t:
Counter example: day of week, time of day

5. Template Transition Model
Weather
Weather’
Velocity
Velocity’
Location
Location’
Failure
Failure’
Obs’
Time slice t
Time slice t+1

6. Initial State Distribution
Weather0
Velocity0
Location0
Failure0
Obs0
Time slice 0

7. Ground Bayesian Network
Weather0
Location1
Failure1
Obs1
Time slice 1
Velocity1
Weather1
Location2
Failure2
Obs2
Velocity2
Weather2
Time slice 2
Velocity0
Location0
Failure0
Obs0
Time slice 0

8. 2-time-slice Bayesian Network
A transition model (2TBN) over X1,…,Xn is specified as a BN fragment such that:
The nodes include X1’,…,Xn’ and a subset of X1,…,Xn
Only the nodes X1’,…,Xn’ have parents and a CPD
The 2TBN defines a conditional distribution

9. Dynamic Bayesian Network
A dynamic Bayesian network (DBN) over X1,…,Xn is defined by a
2TBN BN over X1,…,Xn
a Bayesian network BN(0) over X1(0) ,…,Xn(0)

10. Ground Network
For a trajectory over 0,…,T we define a ground (unrolled network) such that
The dependency model for X1(0) ,…,Xn(0) is copied from BN(0)
The dependency model for X1(t) ,…,Xn(t) for all t > 0 is copied from BN

11. Hidden Markov Models s1 s2 s3 s4 S S’ S1 O1 S0 S2 O2 S3 O3 O’ 0.5 0.7
0.3
0.4
0.6
0.1
0.9

12. Consider a smoke detection tracking application, where we have 3 rooms connected in a row. Each room has a true smoke level (X) and a smoke level (Y) measured by a smoke detector situated in the middle of the room. Which of the following is the best DBN structure for this problem?
X’1
Y’1 X1
X’2
Y’2 X2
X’3
Y’3 X3
X’1
Y’1 X1
X’2
Y’2 X2
X’3
Y’3 X3
X’1
Y’1 X1
X’2
Y’2 X2
X’3
Y’3 X3
X’1
Y’1 X1
X’2
Y’2 X2
X’3
Y’3 X3

13. Robot Localization ... u u u x x x x map z z z control signal1
t-1
robot pose
... x x x x 1 2 t
map z z z
sensor observation 1 2 t

14. Tim Huang, Dieter Koller, Jitendra Malik, Gary Ogasawara, Bobby Rao, Stuart Russell, J. Weber

15. Consider an application, where we have 3 sources, each making sound (X) simultaneously (e.g., a TV, a blender, and a ringing cellphone). We want to separate the observed aggregate signal (Y) into its unobserved constituent signals. Which of the following is the best DBN structure for this problem?
X’1 X1
X’2 X2
X’3 Y X3 X1
X’1 X2
X’2 X3
X’3 Y
X’1
Y’1 X1
X’2
Y’2 X2
X’3
Y’3 X3
X’1
Y’1 X1
X’2
Y’2 X2
X’3
Y’3 X3

16. Consider a situation where we have two people in a conversation, each wearing a sensitive lapel microphone. We want to separate each person’s speech (X) using the signal (Y) obtained from their microphone. Which of the following is the best DBN structure for this problem? X1
X’1 X1
X’1
Y’1
Y’1 X2
X’2 X2
X’2
Y’2
Y’2 X1
X’1 X1
X’1
Y’1 X2
X’2
Y’1 X2
X’2
Y’2
Y’2

17. Summary
DBNS are a compact representation for encoding structured distributions over arbitrarily long temporal trajectories
They make assumptions that may require appropriate model (re)design:
Markov assumption
Time invariance