●●°| Probab| listic Mode ● EStimate p(X Be(x,)=p(x,|=,a1,1,a12,=0) ● Bayes’rule Bel(x,= P(,|x,a1-12=1,a12…,2=0)p(x1a1,21,a12,-) tt-1-t-1t-2…:-0 my(1|x1)p(x1an1,x1,a12x,=0)
Bel (x )t Probabilistic Model z Estimate p(xt): Bel (x ) = x p | a z t −1, zt −1, at −2 ,..., z ) t ( t t , 0 z Bayes’ rule: (z p | x , a ( t −1, zt −1, at −2 ,..., z ) x p | at −1, zt −1, at −2 ,..., z ) = t t 0 t (z p t | at −1, zt −1, at −2 ,..., z )0 =α ( ( z p | x ) x p | at −1, zt −1, at −2 ,..., z ) t t t 0 0
●●°| Probab| listic Mode o Integrate over all p(x-1) Bel(x)=∞p(1|x1)p(x1an12E1,a1-2,2-0) (=,|x,)p(x,|x )p(x-1|a12,-0)x1 x-1 D(1x)m(x1|x1,a-)(x)
Probabilistic Model z Integrate over all p(xt-1): Bel (x ) = α ( ( z p | x ) x p | at −1, zt −1, at − 2 ,..., z ) t t t t 0 z p | xt ) ∫ x p t | xt −1, at −1, zt − 2 ,..., z ) x p | at −1 = α ( ( ( ,..., z )dx t 0 t −1 0 t −1 xt −1 z p | xt ) ∫ x p t | xt −1, at −1 = α ( ( ) (x p )dx t t −1 t −1 xt −1
●。。 Probab| istic Mode o Bayes' filter gives recursive, two-step procedure for estimating p(X,) Bel(,)+ap(= l*, p(x, 1x,a p(*-)dr Measurement Prediction o How to represent Bel(X+?
Probabilistic Model | Bayes’ filter gives recursive, two-step procedure for estimating p(x t x t) Bel (x ) = αp (z | x ) p (x | xt − 1, at − 1) p (x )dx t t t ∫ t t −1 t − 1 −1 Measurement Prediction | How to represent Bel(xt)?
Kalman, 1960 ●。。 Kalman filter An action is taken State space Posterior belier Initial belier Posterior belief after sensing after an action
Kalman, 1960 An action is taken Kalman Filter State Space Posterior belief Posterior belief Initial belief after an action after sensing
●。 Problems o Gaussian process and sensor noise Often solved extracting low-dimensional features Data-association problem o Often hard to implement Kalman filters o Gaussian posterior estimate
Problems | Gaussian Process and Sensor Noise • Often solved extracting low-dimensional features • Data-association problem | Often hard to implement Kalman filters | Gaussian Posterior Estimate