  Motivated by the recent experimental observation of a large pressure effect
on $T_c$ in Pr$_{1-x}$Sr$_x$NiO$_2$, we study the electronic properties of this
compound as a function of pressure for $x=0$ and $0.2$ doping using
self-consistent $GW$+EDMFT. Our numerical results demonstrate a non-trivial
interplay between chemical doping and physical pressure, and small but
systematic changes in the orbital occupations, local level energies, and
interaction parameters with increasing pressure. The proper treatment of
correlation effects, beyond density function theory, is shown to play an
important role in revealing these trends. While the pressure dependent changes
in the electronic structure of the undoped compound suggest a more
single-band-like behavior in the high-pressure regime, a qualitatively
different behavior is found in the doped system. We also point out that the
fluctuations in the orbital occupations and spin states are not consistent with
a single-band picture, and that at least a two-band model is necessary to
reproduce the full result. This multi-orbital nature manifests itself most
clearly in the doped compound.
