  We introduce statistical, conjugate connection and Hessian structures on
anti-commutable pre-Leibniz algebroids. Anti-commutable pre-Leibniz algebroids
are special cases of local pre-Leibniz algebroids, which are still general
enough to include many physically motivated algebroids such as Lie, Courant,
metric and higher-Courant algebroids. They create a natural framework for
generalizations of differential geometric structures on a smooth manifold. The
symmetrization of the bracket on an anti-commutable pre-Leibniz algebroid
satisfies a certain property depending on a choice of an equivalence class of
connections which are called admissible. These admissible connections are shown
to be necessary to generalize aforementioned structures on pre-Leibniz
algebroids. Consequently, we prove that, provided certain conditions,
statistical and conjugate connection structures are equivalent when defined for
admissible connections. Moreover, we also show that for
`projected-torsion-free' connections, one can generalize Hessian metrics and
Hessian structures. We prove that any Hessian structure yields a statistical
structure, where these results are completely parallel to the ones in the
manifold setting. We also prove a mild generalization of the fundamental
theorem of statistical geometry. Moreover, we generalize $\alpha$-connections,
strongly conjugate connections and relative torsion operator, and prove some
analogous results.
