Probabilistic Graphical Models Representation Template Models Temporal Models

Probabilistic Graphical Models Representation Template Models Temporal Models Daphne Koller

Distributions over Trajectories • Pick time granularity • X(t) – variable X at time t • X(t: t’) = {X(t), …, X(t’)} (t t’) 0 1 2 3 4 5 • Want to represent P(X(t: t’)) for any t, t’ Daphne Koller

Markov Assumption Daphne Koller

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

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

Initial State Distribution Weather 0 Velocity 0 Location 0 Failure 0 Obs 0 Time slice 0 Daphne Koller

Ground Bayesian Network Weather 0 Weather 1 Weather 2 Velocity 0 Velocity 1 Velocity 2 Location 0 Location 1 Location 2 Failure 0 Failure 1 Failure 2 Obs 0 Time slice 0 Obs 1 Time slice 1 Obs 2 Time slice 2 Daphne Koller

2 -time-slice Bayesian Network • A transition model (2 TBN) over X 1, …, Xn is specified as a BN fragment such that: – The nodes include X 1’, …, Xn’ and a subset of X 1, …, Xn – Only the nodes X 1’, …, Xn’ have parents and a CPD • The 2 TBN defines a conditional distribution Daphne Koller

Dynamic Bayesian Network • A dynamic Bayesian network (DBN) over X 1, …, Xn is defined by a – 2 TBN BN over X 1, …, Xn – a Bayesian network BN(0) over X 1(0) , …, Xn(0) Daphne Koller

Ground Network • For a trajectory over 0, …, T we define a ground (unrolled network) such that – The dependency model for X 1(0) , …, Xn(0) is copied from BN(0) – The dependency model for X 1(t) , …, Xn(t) for all t > 0 is copied from BN Daphne Koller

Hidden Markov Models S S’ S 0 O’ 0. 3 s 1 s 2 S 3 O 1 O 2 O 3 0. 1 0. 5 0. 7 S 1 0. 4 0. 6 s 3 0. 5 s 4 0. 9 Daphne Koller

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 X 2 X’ 1 X’ 3 X 1 X’ 1 X 3 Y’ 1 X’ 2 X 3 X 2 X 1 X’ 2 X’ 3 Y’ 2 Y’ 3 Y’ 1 Y’ 2 Y’ 3 X 2 X’ 1 X’ 2 X 3 X’ 3 X 1 X’ 1 X 2 X 3 X’ 2 X’ 3 Y’ 1 Y’ 2 Y’ 3 Daphne Koller

Robot Localization u 0 u 1 ut-1 control signal robot pose x 0 x 1 x 2 z 1 z 2 . . . xt map sensor observation zt Daphne Koller

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

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 Daphne Koller