  Observational entropy provides a general notion of quantum entropy that
appropriately interpolates between Boltzmann's and Gibbs' entropies, and has
recently been argued to provide a useful measure of out-of-equilibrium
thermodynamic entropy. Here we study the mathematical properties of
observational entropy from an information-theoretic viewpoint, making use of
recently strengthened forms of the monotonicity property of quantum relative
entropy. We present new bounds on observational entropy applying in general, as
well as bounds and identities related to sequential and post-processed
measurements. A central role in this work is played by what we call the
``coarse-grained'' state, which emerges from the measurement's statistics by
Bayesian retrodiction, without presuming any knowledge about the ``true''
underlying state being measured. The degree of distinguishability between such
a coarse-grained state and the true (but generally unobservable) one is shown
to provide upper and lower bounds on the difference between observational and
von Neumann entropies.
