  This work is about low dimensional reduction for a slow-fast data
assimilation system with non-Gaussian $\alpha-$stable L\'evy noise via
stochastic averaging. When the observations are only available for slow
components, we show that the averaged, low dimensional filter approximates the
original filter, by examining the corresponding Zakai stochastic partial
differential equations. Furthermore, we demonstrate that the low dimensional
slow system approximates the slow dynamics of the original system, by examining
parameter estimation and most probable paths.
