Filter stability is a classical problem for partially observed Markov processes (POMP). For a POMP, 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 either focus on the ergodic properties on the hidden Markov process, or the informativeness/observability properties of the measurement channel. While notions of observability exist in the literature, they are often difficult to verify and specific examples of observable systems are mostly restricted to additive noise models with additional strict regularity properties. In this paper, we introduce a general definition of observability for stochastic non-linear dynamical systems and compare it with related findings in the literature. Our observability notion involves a functional characterization which is easily computed for a variety of systems as we demonstrate. Under this observability definition we establish filter stability results for a variety of criteria including weak merging and total variation merging, both in expectation and in an almost sure sense, as well as relative entropy. We consider the implications between these notions, which unify various results in the literature in a concise manner. Our conditions, and the examples we study, complement and generalize the existing results on filter stability.

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