Graphical models, bayes' rule and the markov Assumption actions Beliefs T(xi a Observations (Z, Observable O(z /x) states Bayes rule: P(x y) p(y)p p(y) Markov:p(x1|x1,a1,a0,z0,a1,21,…,z)=p(x1|x1,a1) The Bayes' Filter p, (x)=p(x ao, zo, a,, z, a ,,z, According to Bayes rule: p(z|x,a0,z0,a1,z1,a2,2…,a,)p(xao,20,a1,2,a a.) According to the markov assumption p(z,)p(x Introducing an auxiliary variable p(z,1x)P(x|a, x')P(x ao, Zo, a, -,,a-1, 2-1) And with recursion =p(z, 1x)p(xla, x')pi-I(x)Graphical Models, Bayes’ Rule and the Markov Assumption States x1 x2 T(xj |ai , xi ) Z2 b Beliefs 1 Observations Z1 a Actions 1 O(zj |xi ) b2 Z2 Hidden Observable ( ) ( | ) ( ) ( | ) p y p y x p x Bayes rule : p x y ( | , , , , , , , ) ( | , ) t t 1 t 0 0 1 1 t 1 t t 1 at p x x a a z a z z p x x     Markov : The Bayes’ Filter ( | ) ( | , ') ( ) ( | ) ( | , ') ( '| , , , ,..., , ) ( | ) ( | , , , , , ,..., ) ( | , , , , , , ,..., ) ( | , , , , , ,..., ) ( ) ( | , , , , , ,..., , ) 1 ' 0 0 1 1 1 1 ' 0 0 1 1 2 2 0 0 1 1 2 2 0 0 1 1 2 2 0 0 1 1 2 2 p z x p x a x p x p z x p x a x p x a z a z a z p z x p x a z a z a z a p z x a z a z a z a p x a z a z a z a p x p x a z a z a z a z t X t t t t X t t t t t t t t t t    ³ ³ And with recursion: Introduc ing an auxiliary variable : Accordi ng to the Markov assumption : Accordi ng toBayes rule :