  In this paper, we study the problem of estimating a Markov chain $X$(signal)
from its noisy partial information $Y$, when the transition probability kernel
depends on some unknown parameters. Our goal is to compute the conditional
distribution process ${\mathbb P}\{X_n|Y_n,...,Y_1\}$, referred to hereafter as
the {\it optimal filter}. Following a standard Bayesian technique, we treat the
parameters as a non-dynamic component of the Markov chain. As a result, the new
Markov chain is not going to be mixing, even if the original one is. We show
that, under certain conditions, the optimal filters are still going to be
asymptotically stable with respect to the initial conditions. Thus, by
computing the optimal filter of the new system, we can estimate the signal
adaptively.
