Stability of Non-Linear Filters, Observability and Relative Entropy

For a partially observed Markov chain, an incorrectly initialized non-linear filter is said to be stable if the filter eventually corrects itself with the arrival of new measurement information. In the literature, studies on the stability of non-linear filters have either assumed strong stationary conditions on the hidden Markov process, or have assumed restrictive assumptions on the observation channel structure. In this paper, compared to existing results in the literature, on the one hand we relax the assumptions on the observation channel and on the other hand, by using relative entropy as a versatile tool for proving convergence and stability, we are able to obtain complementary convergence results under observability conditions.