  By performing accurate ab-initio density functional theory calculations, we
study the role of $4f$ electrons in stabilizing the magnetic-field-induced
ferroelectric state of DyFeO$_{3}$. We confirm that the ferroelectric
polarization is driven by an exchange-strictive mechanism, working between
adjacent spin-polarized Fe and Dy layers, as suggested by Y. Tokunaga [Phys.
Rev. Lett, \textbf{101}, 097205 (2008)]. A careful electronic structure
analysis suggests that coupling between Dy and Fe spin sublattices is mediated
by Dy-$d$ and O-$2p$ hybridization. Our results are robust with respect to the
different computational schemes used for $d$ and $f$ localized states, such as
the DFT+$U$ method, the Heyd-Scuseria-Ernzerhof (HSE) hybrid functional and the
GW approach. Our findings indicate that the interaction between the $f$ and $d$
sublattice might be used to tailor ferroelectric and magnetic properties of
multiferroic compounds.
