  We demonstrate that the proper general setting for contrast (potential)
functions in statistical and information geometry is the one provided by Lie
groupoids and Lie algebroids. The contrast functions are defined on Lie
groupoids and give rise to two-forms and three-forms on the corresponding Lie
algebroid. If the two-form is non-degenerate, it defines a `pseudo-Riemannian'
metric on the Lie algebroid and a family of Lie algebroid torsion-free
connections, including the Levi-Civita connection of the metric. In this
framework, the two-point functions are just functions on the pair groupoid
$M\ti M$ with the `standard' metric and affine connection on the Lie algebroid
$\sT M$. We study also reductions of such systems and infinite-dimensional
examples. In particular, we find a contrast function defining the Fubini-Study
metric on the Hilbert projective space.
