  Filter stability is a classical problem in the study of partially observed
Markov processes (POMP), also known as hidden Markov models (HMM). For a POMP,
an incorrectly initialized non-linear filter is said to be (asymptotically)
stable if the filter eventually corrects itself as more measurements are
collected. Filter stability results in the literature that provide rates of
convergence typically rely on very restrictive mixing conditions on the
transition kernel and measurement kernel pair, and do not consider their
effects independently. In this paper, we introduce an alternative approach
using the Dobrushin coefficients associated with both the transition kernel as
well as the measurement channel. Such a joint study, which seems to have been
unexplored, leads to a concise analysis that can be applied to more general
system models under relaxed conditions: in particular, we show that if $(1 -
\delta(T))(2-\delta(Q)) < 1$, where $\delta(T)$ and $\delta(Q)$ are the
Dobrushin coefficients for the transition and the measurement kernels, then the
filter is exponentially stable. Our findings are also applicable for controlled
models.

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