  We develop algebraic tools for statistical inference from samples of rotation
matrices. This rests on the theory of D-modules in algebraic analysis.
Noncommutative Gr\"obner bases are used to design numerical algorithms for
maximum likelihood estimation, building on the holonomic gradient method of
Sei, Shibata, Takemura, Ohara, and Takayama. We study the Fisher model for
sampling from rotation matrices, and we apply our algorithms for data from the
applied sciences. On the theoretical side, we generalize the underlying
equivariant D-modules from SO(3) to arbitrary Lie groups. For compact groups,
our D-ideals encode the normalizing constant of the Fisher model.
