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Tracking on the rotation group is a key component of many modern systems for estimation of the motion of rigid bodies. To address this problem, here we describe a Bayesian algorithm that relies on directional measurements for tracking on the special orthogonal (rotation) group. Its novelty lies in the use of maximum entropy distributions on these groups as models for the priors, and justifiable approximation algorithms that permit recursive implementation of such a model. We provide the solutions in a recursive closed form. In the two-dimensional case the parameters of the prior and posterior distributions can be computed exactly and the solution has low complexity. Adoption of this approach eliminates the problem of angle wrapping. In higher dimensions the exact solution cannot be computed, and it is necessary to make (very close) approximations, which is done here. We demonstrate in simulations that, in contrast with some other approaches, our algorithm produces very accurate and statistically meaningful outputs.