  We present schemes for simulating Brownian bridges on complete and connected
Lie groups and homogeneous spaces. We use this to construct an estimation
scheme for recovering an unknown left- or right-invariant Riemannian metric on
the Lie group from samples. We subsequently show how pushing forward the
distributions generated by Brownian motions on the group results in
distributions on homogeneous spaces that exhibit non-trivial covariance
structure. The pushforward measure gives rise to new parametric families of
distributions on commonly occurring spaces such as spheres and symmetric
positive tensors. We extend the estimation scheme to fit these distributions to
homogeneous space-valued data. We demonstrate both the simulation schemes and
estimation procedures on Lie groups and homogenous spaces, including $\SPD(3) =
\GL_+(3)/\SO(3)$ and $\mathbb S^2 = \SO(3)/\SO(2)$.
