  We consider some natural (functorial) lifts of geometric objects associated
with statistical manifolds (metric tensor, dual connections, skewness tensor,
etc.) to higher tangent bundles. It turns out that the lifted objects form
again a statistical manifold structure, this time on the higher tangent
bundles, with the only difference that the metric tensor is pseudo-Riemannian.
What is more, natural lifts of potentials (called also divergence or contrast
functions) turn out to be again potentials, this time for the lifted
statistical structures. We propose an analogous procedure for lifting
statistical structures on Lie algebroids and lifting contrast functions which
are defined on Lie groupoids. In particular, we study in detail Lie groupoid
structures of higher tangent bundles of Lie groupoids. Our geometric
constructions of lifts are illustrated by explicit examples, including some
important statistical models and potential functions on Lie groupoids.
