  An open quantum system in contact with an infinite bath approaches
equilibrium, while the state of the bath remains unchanged. If the bath is
finite, the open system still relaxes to equilibrium, but it induces a
dynamical evolution of the bath state. In this work, we study the dynamics of
open quantum systems in contact with finite baths. We obtain a hierarchy of
master equations that improve their accuracy by including more dynamical
information of the bath. For instance, as the least accurate but simplest
description in the hierarchy we obtain the conventional Born-Markov-secular
master equation. Remarkably, our framework works even if the measurements of
the bath energy are imperfect, which, not only is more realistic, but also
unifies the theoretical description. Also, we discuss this formalism in detail
for a particular non-interacting environment where the Boltzmann temperature
and the Kubo-Martin-Schwinger relation naturally arise. Finally, we apply our
hierarchy of master equations to study the central spin model.
