  Nakajima's graded quiver varieties naturally appear in the study of bases of
cluster algebras. One particular family of these varieties, namely the
bipartite determinantal varieties, can be defined for any bipartite quiver and
gives a vast generalization of classical determinantal varieties with broad
applications to algebra, geometry, combinatorics, and statistics. The ideals
that define bipartite determinantal varieties are called bipartite
determinantal ideals.
  We provide an elementary method of proof showing that the natural generators
of a bipartite determinantal ideal form a Gr\"obner basis, using an
S-polynomial construction method that relies on the Leibniz formula for
determinants. This method is developed from an idea by Narasimhan and
Caniglia--Guccione--Guccione.
  As applications, we study the connection between double determinantal ideals
(which are bipartite determinantal ideals of a quiver with two vertices) and
tensors, and provide an interpretation of these ideals within the context of
algebraic statistics.
