Part 6

• State-Observation Models as Distributions

  A) Joint Distribution

  B) i

  C) i-1

  D) O^i

  E) S^i

  F) S^0

  
• Three common inference tasks in S-O-Models

  Filtering/tracking

  Smoothing

  Prediction
• Filtering/tracking


  Compute our belief about the current system state given all observations so far
• Smoothing 


  Compute the posterior distribution over the system state at time t given all of the evidence over some longer trajectory
• Prediction : given all the observation up to t, predict the distribution over some future states
• 3 Common Inference Tasks in S-O-Models

  A) Filtering/tracking

  B) Smoothing

  C) Prediction

  
• Hindsight : Later information may change our belief about previous states that the system might have gone through
• Filtering problems :

  sum out over a large number of hidden variables

  must keep entire histroy of observations
• Exact inference in S-O Models

  A) 4x100+3

  
• Recursive inductive update algorithm :

  propagate state forward

  take into account new observation
• Posterior Belief State : $P(S^{(t+1)} | o^{(1:t+1)}$
• The general filtering algorithm

  A) Propagate state distribution forwards

  B) Condition on new observation

  
• Forward step

  A) initialisation

  B) state propagation

  C) Conditioning and re-normalisation

  
• Prediction (converge to the stationary distribution of the Markov process)

  A) Transition Model

  
• Trick to smoothing is to devide the task into two parts - the evidence up to k and the evidence from k+1 to T
• Smoothing

  A) forward message

  B) backward message

  C) Forward-Backward Algortihm

  
• The forward backward algorithm

  A) Initialise

  B) forward pass

  C) store

  D) Initialise b

  E) Backward pass

  
• Forward pass

  go forward in time, estimate probability distribution over current state, given observations so far
• Backward pass


  go backward in time, in each step correct forward estimate by considering what happened later
• Summary over general problems

  A) intractable

  B) exponential

  C) sum

  D) possible states