P ( z t | x t ) P ( x t | u t , x t 1 ) Bel ( x t 1 ) dx t 1 Bayes Filters Bayes z = observation u = action x = state P ( x t | u 1 , z 1 … , u t , z t ) P ( z t | x t , u 1 , z 1 , … , u t ) P ( x t | u 1 , z 1 , … , u t ) P ( z t | x t ) P ( x t | u 1 , z 1 , … , u t ) 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 P ( z t | x t ) P ( x t | u t , x t 1 ) P ( x t 1 | u 1 , z 1 , … , u t ) dx t 1 P ( z t | x t ) P ( x t | u t , x t 1 ) P ( x t 1 | u 1 , z 1 , … , z t 1 ) dx t 1 Bel ( x t ) Mark o v Mark o v Total prob. Mark o v