Recursive computations for the optimal tracking of time-varying parameters (Corresp.)

Recursive relations are given for updating the conditional density
p(\theta_{k} | X_{k-1}, \cdots X_{1})
(also for
p(\theta_{k} | X_{k}, \cdots, X_{1})
), where
\theta_{k}
is a parameter of the density of
X_{k}
. The observations
X_{1}, X_{2}, \cdots
are assumed to be conditionally independent (i.e., for known parameters), and the sequence of time-varying parameters
\theta_{1}, \theta_{2}, \cdots
constitutes a Markov-M sequence. The result requires the storage of an intermediate function of
(\theta_{k-1}, \cdots , \theta_{k-M})
.