  We present a new strategy for filtering high-dimensional multiscale systems
characterized by high-order non-Gaussian statistics using observations from
leading-order moments. A closed stochastic-statistical modeling framework
suitable for systematic theoretical analysis and efficient numerical
simulations is designed. Optimal filtering solutions are derived based on the
explicit coupling structures of stochastic and statistical equations subject to
linear operators, which satisfy an infinite-dimensional Kalman-Bucy filter with
conditional Gaussian dynamics. To facilitate practical implementation, we
develop a finite-dimensional stochastic filter model that approximates the
optimal filter solution. We prove that this approximating filter effectively
captures key non-Gaussian features, demonstrating consistent statistics with
the optimal filter first in its analysis step update, then at the long-time
limit guaranteeing stable convergence to the optimal filter. Finally, we build
a practical ensemble filter algorithm based on the approximating filtering
model, which enables accurate recovery of the true model statistics. The
proposed modeling and filtering strategies are applicable to a wide range
challenging problems in science and engineering, particularly for statistical
prediction and uncertainty quantification of multiscale turbulent states.
