Bayesian filtering techniques: Kalman and extended Kalman filter basics

Bayesian filters provide a statistical tool for dealing with measurement uncertainty. Bayesian filters estimate a state of dynamic system from noisy observations. These filters represent the state by random variable and in each time step probability distribution over random variable represents the uncertainty. If estimate is needed with every new measurement, it is suitable to use recursive filter. Unfortunately optimal Bayesian solution exists in a restrictive set of cases, e.g. Kalman filters which assume Gaussian PDF or we need to use suboptimal solution, e.g. extended Kalman filters which use local linearization to approximate PDF to be Gaussian.