Temporal Models Template Models Representation Probabilistic Graphical

Distributions over Trajectories 1 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’

Markov Assumption Counter example: location and velocity

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

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

Initial State Distribution Weather0 Velocity0 Location0 Failure0 Obs0 Time slice 0

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

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

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)

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

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

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

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

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