  We discuss the geometric foundation behind the use of stochastic processes in
the frame bundle of a smooth manifold to build stochastic models with
applications in statistical analysis of non-linear data. The transition
densities for the projection to the manifold of Brownian motions developed in
the frame bundle lead to a family of probability distributions on the manifold.
We explain how data mean and covariance can be interpreted as points in the
frame bundle or, more precisely, in the bundle of symmetric positive definite
2-tensors analogously to the parameters describing Euclidean normal
distributions. We discuss a factorization of the frame bundle projection map
through this bundle, the natural sub-Riemannian structure of the frame bundle,
the effect of holonomy, and the existence of subbundles where the Hormander
condition is satisfied such that the Brownian motions have smooth transition
densities. We identify the most probable paths for the underlying Euclidean
Brownian motion and discuss small time asymptotics of the transition densities
on the manifold. The geometric setup yields an intrinsic approach to the
estimation of mean and covariance in non-linear spaces.
