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Bayes Filters: Framework GP-Based WiFi Sensor Model CSE-571 Given: Stream of observations z and action data u: Robotics = d { u , z ! , u , z } - t 1 2 t 1 t Sensor model P(z|x). Action model P(x|u,x ) .

  1. Bayes Filters: Framework GP-Based WiFi Sensor Model CSE-571 • Given: • Stream of observations z and action data u: Robotics = d { u , z ! , u , z } - t 1 2 t 1 t • Sensor model P(z|x). • Action model P(x|u,x � ) . Bayes Filters Mean • Prior probability of the system state P(x). • Wanted: • Estimate of the state X of a dynamical system. • The posterior of the state is also called Belief : = ! Bel ( x ) P ( x | u , z , u , z ) t t 1 2 t - 1 t Variance CSE-571: Probabilistic 10/6/16 Robotics 2 SA-1 z = observation Bayes Filter Algorithm ò u = action = h Bayes Filters Bel ( x ) P ( z | x ) P ( x | u , x ) Bel ( x ) dx x = state - - - t t t t t t 1 t 1 t 1 Algorithm Bayes_filter ( Bel(x),d ): = 1. Bel ( x ) P ( x | u , z ! , u , z ) t t 1 1 t t 2. n = 0 = h P ( z | x , u , z , ! , u ) P ( x | u , z , ! , u ) 3. If d is a perceptual data item z then Bayes t t 1 1 t t 1 1 t 4. For all x do = h P ( z | x ) P ( x | u , z , ! , u ) = Bel ' ( x ) P ( z | x ) Bel ( x ) Markov 5. t t t 1 1 t h = h + Bel ' x ( ) 6. Total prob. 7. For all x do ∫ = η P ( z t | x t ) P ( x t | u 1 , z 1 , … , u t , x t − 1 ) P ( x t − 1 | u 1 , z 1 , … , u t ) dx t − 1 = h - 1 8. Bel ' ( x ) Bel ' ( x ) 9. Else if d is an action data item u then ò = h P ( z | x ) P ( x | u , x ) P ( x | u , z , ! , u ) dx Markov t t t t t - 1 t - 1 1 1 t t - 1 10. For all x do ∫ Bel '( x ) = 11. P ( x | u , x ') Bel ( x ') dx ' ò = h P ( z | x ) P ( x | u , x ) Bel ( x ) dx - - - 12. Return Bel ’ (x) t t t t t 1 t 1 t 1 1

  2. Markov Assumption Dynamic Environments • Two possible locations x 1 and x 2 • P(x 1 )=0.99 • P(z| x 2 )=0.09 P(z| x 1 )=0.07 1 p(x2 | d) p(x1 | d) 0.9 = p ( z | x , z , u ) p ( z | x ) 0.8 - t 0 : t 1 : t 1 1 : t t t 0.7 = p ( x | x , z , u ) p ( x | x , u ) 0.6 - - - p( x | d) t 1 : t 1 1 : t 1 1 : t t t 1 t 0.5 0.4 Underlying Assumptions 0.3 • Static world 0.2 0.1 • Independent noise 0 5 10 15 20 25 30 35 40 45 50 Number of integrations • Perfect model, no approximation errors Representations for Bayesian Robot Localization Kalman filters (late-80s) • Gaussians, unimodal Discrete approaches ( ’ 95) • approximately linear models • Topological representation ( ’ 95) • position tracking • uncertainty handling (POMDPs) • occas. global localization, recovery Bayes Filters for • Grid-based, metric representation ( ’ 96) Robot Localization • global localization, recovery Robotics AI Particle filters ( ’ 99) Multi-hypothesis ( ’ 00) • sample-based representation • multiple Kalman filters • global localization, recovery • global localization, recovery 2

  3. Bayes Filters are Familiar! Summary • Bayes rule allows us to compute ò = h Bel ( x ) P ( z | x ) P ( x | u , x ) Bel ( x ) dx - - - t t t t t t 1 t 1 t 1 probabilities that are hard to assess otherwise. • Kalman filters • Under the Markov assumption, • Particle filters recursive Bayesian updating can be • Hidden Markov models used to efficiently combine evidence. • Dynamic Bayesian networks • Bayes filters are a probabilistic tool • Partially Observable Markov Decision Processes (POMDPs) for estimating the state of dynamic systems. 3

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