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

Cumulants, Koszul brackets and homological perturbation theory for commutative $BV_\infty$ and $IBL_\infty$ algebras