Relative entropy and error bounds for filtering of Markov processes

This paper considers the relative entropy between the conditional distribution and an incorrectly initialized filter far the estimation of one component of a Markov process given observations of the second component. Using the Markov property ae first establish a decomposition of the relative entropy between the measures on observation path space associated to different initial conditions. Using this decomposition, it is shown that the relative entropy of the optimal filter relative to an incorrectly initialized filter is a positive supermartingale. By applying the decomposition to signals observed in additive, white noise, a relative entropy bound is obtained on the integrated, expected, mean square difference between the optimal and incorrectly initialized estimates of the observation function.