  The main purpose of the present work is to investigate statistical manifolds
endowed with almost product structures. We prove that the statistical structure
of a para-K\"{a}hler-like statistical manifold of constant curvature in the
Kurose's sense is a Hessian structure. We also derive the main properties of
statistical submersions which are compatible with almost product structures.
The results are illustrated by several nontrivial examples.
