For a smooth manifold M, the tangent groupoid “glues” the set \(M \times M\) with TM as two underlying pieces in smooth transition from one to the other. We show that any contrast function defined on \(M \times M\) naturally leads to a Riemannian metric and a pair of torsion-free conjugate connections (so-called “statistical structure”) that are objects defined for sections of TM. This is achieved through smooth “extension” of the contrast function and its anti-symmetrized version on \(M \times M\) to, respectively, a quadratic and a cubic function on TM. We recovered the standard formulae [1, 4–6] linking contrast functions to statistical structure through differentiation of the former by two and three vector fields to obtain the metric and the connections, respectively.