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The state estimation problem for bilinear stochastic systems evolving on the spheres Sn, the special orthogonal groups , and certain other compact Lie groups and homogeneous spaces is considered. The problem is motivated by some applications involving rotational processes in three dimensions. The theory of harmonic analysis on compact Lie groups is used to define assumed density approximations which result in implementable suboptimal estimators for the state of the bilinear system. The results of Monte Carlo simulations are reported; these indicate that simple filters designed by these techniques perform well as compared to other filters.