Computational Logics

• System "patient" is a dynamic system whose state changes over time and whose state we wish to track and reason about :

  • variable have different values at different times
  • time t may depend on previous sttes
• Dynamic Bayesian Networks (DBNs)

  models of temporal processes

  development over time

  model distributions over sequences of system states
• $X^{(t)}_{i}$ means each variable $X_i$ has a specific instantiation for each t
• the mother of all variables $X^{(t)}_{i}$ : $X_i$template variable
• BDNs complexity problem :

  trajector : an assigmnet of values to all variables for some duration of T

  joint distribution over such trajectories
• Simplifying assumption 1 : Discrete Time :

  timeline is discretised into time slices

  step size delta

  consequence :

  finite (large) set of random variabels

  trajectory distribution (image)

  A) t+1

  B) 0:t

  
• Simplifying assumption 2 : The markov assumption

  the future is conditionally independant of the past, given the present

  -

  very strong limitation

  not satisfied in a lot of application

  A) t+1

  B) 0:t-1

  C) independent

  
• The Markov Assumption

  A) T-1

  B) t+1

  C) t

  
• Simplifying assumption 3 : Stationarity

  laws governing the system' s behaviour do not change over time

  are the same in each time steps

  stationary dynamics
• A Markovian dynamic system is stationary (or time invariant) if $P(X^{t+1} | X^t)$ is the same for all t
• Consequence of stationary :

  the intial state distribution

  the transition model $P(X' | X)$
• Dynamic bayesian network (DBN)

  
• Dynamic Bayesian Network is a pair where

  Bayesian network over $X^{(0)}$ -> distribution over the intial states

  two-timeslice network that describes the transition model $P(X'|X)$
• the probability distribution over the trajectors is defined by an unrolled bayesian network
• Given all the sensor readings, from the beginning and the weater(t) where is the car now?

  
• Given the cars current velocity and the current sensor reading, where will the car be in 2 seconds and what will be the speed .

  
• Problems with inference in DBN

  inference intractable

  may want to perform online reasoning
• Exact inference in unconstrained DBN is computationally extremely expensive/intracable
• State-Observation Models : split variables into two subsets : State Variables S (unobservable) and Observation Variables O

  
• A State-Observation Model is a DBN that consists of three compotents :

  1. an intital state model
  2. a state transition model
  3. an observation model
• A State-Observation Model is a DBN that consists of three compotents :

  an intital state model $P(S^{(0)})$

  a state transition model $P(S' | S)$

  an observation model $P(O|S)$
• State-Observation Models :

  state transitions satisfy the Markov assumption

  the current observations depend only on the current state