  We revisit the development of grid based recursive approximate filtering of
general Markov processes in discrete time, partially observed in conditionally
Gaussian noise. The grid based filters considered rely on two types of state
quantization: The \textit{Markovian} type and the \textit{marginal} type. We
propose a set of novel, relaxed sufficient conditions, ensuring strong and
fully characterized pathwise convergence of these filters to the respective
MMSE state estimator. In particular, for marginal state quantizations, we
introduce the notion of \textit{conditional regularity of stochastic kernels},
which, to the best of our knowledge, constitutes the most relaxed condition
proposed, under which asymptotic optimality of the respective grid based
filters is guaranteed. Further, we extend our convergence results, including
filtering of bounded and continuous functionals of the state, as well as
recursive approximate state prediction. For both Markovian and marginal
quantizations, the whole development of the respective grid based filters
relies more on linear-algebraic techniques and less on measure theoretic
arguments, making the presentation considerably shorter and technically
simpler.
