1. Temporal Processes Eran Segal Weizmann Institute

2. Representing Time Add the time dimension to variables X  X (t) Assumptions Time can be discretized into interesting points t 1,t 2,...t n Markov assumption holds Distribution is stationary (time invariant or homogeneous)

3. Dynamic Bayesian Networks Pair G 0 and G  such that G 0 is a Bayesian network over X(0) G  is a conditional Bayesian network for P(X(t+1)|X(t)) Weather Velocity Location Failure Weather’ Velocity’ Location’ Failure’ Obs’. GG Weather 0 Velocity 0 Location 0 Failure 0 Obs. 0 G0G0 Weather 0 Velocity 0 Location 0 Failure 0 Weather 1 Velocity 1 Location 1 Failure 1 Obs. 1 Weather 2 Velocity 2 Location 2 Failure 2 Obs. 2 Unrolled network

4. Hidden Markov Model Special case of Dynamic Bayesian network Single (hidden) state variable Single (observed) observation variable Transition probability P(S’|S) assumed to be sparse Usually encoded by a state transition graph SS’ O’ GG G0G0 Unrolled network S0S0 O0O0 S0S0 S1S1 O1O1 S2S2 O2O2 S3S3 O3O3

5. Hidden Markov Model Special case of Dynamic Bayesian network Single (hidden) state variable Single (observed) observation variable Transition probability P(S’|S) assumed to be sparse Usually encoded by a state transition graph S1S1 S2S2 S3S3 S4S4 s1s1 s2s2 s3s3 s4s4 s1s1 0.20.800 s2s2 0010 s3s3 0.4000.6 s4s4 00.50 P(S’|S) State transition representation

6. Inference Forward backward algorithm Forward step: choose last node as root and use BP Backward step: outward messages from last node as root Efficient for HMM, inefficient for DBNs Weather 1 Velocity 1 Location 1 Failure 1 Obs. 1 Weather 2 Velocity 2 Location 2 Failure 2 Obs. 2 Unrolled network Velocity 0 Weather 0 Location 0 Failure 0 There is an active path between any pair of variables in time 2