  We explore the relationship between the classical constructions of cumulants
and Koszul brackets, showing that the former are an expontial version of the
latter. Moreover, under some additional technical assumptions, we prove that
both constructions are compatible with standard homological perturbation theory
in an appropriate sense. As an application of these results, we provide new
proofs for the homotopy transfer Theorem for $L_\infty$ and $IBL_\infty$
algebras based on the symmetrized tensor trick and the standard perturbation
Lemma, as in the usual approach for $A_\infty$ algebras. Moreover, we prove a
homotopy transfer Theorem for commutative $BV_\infty$ algebras in the sense of
Kravchenko which appears to be new. Along the way, we introduce a new
definition of morphism between commutative $BV_\infty$ algebras.
