  In this article, we consider the implications of unobservable subspaces in
the construction of a Kalman filter. In particular, we consider dynamical
systems which are invariant with respect to a group action, and which are
therefore unobservable in the group direction. We obtain reduced propagation
and measurement equations that are invariant with respect to the group action,
and we decompose the state space into unobservable and observable parts. Based
on the decomposition, we propose a reduced Bayesian inference method, which
exhibits superior accuracy for orientation and position estimation, and that is
more robust to large measurement noise.
