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Temporal Models

Representation. Probabilistic Graphical Models. Template Models. Temporal Models. Distributions over Trajectories. 0. 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’.

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Temporal Models

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  1. Representation Probabilistic Graphical Models Template Models TemporalModels

  2. Distributions over Trajectories 0 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’

  3. Markov Assumption

  4. Time Invariance • Template probability model P(X’ | X) • For all t:

  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 Weather1 Weather2 Velocity0 Velocity1 Velocity2 Location0 Location1 Location2 Failure0 Failure1 Failure2 Obs0 Obs1 Obs2 Time slice 2 Time slice 0 Time slice 1

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

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

  13. 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? X1 X1 X1 X1 X’1 X’1 X’1 X’1 Y’1 Y’1 Y’1 Y’1 X2 X2 X2 X2 X’2 X’2 X’2 X’2 Y’2 Y’2 Y’2 Y’2 X3 X3 X3 X3 X’3 X’3 X’3 X’3 Y’3 Y’3 Y’3 Y’3

  14. 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

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