  Observational entropy captures both the intrinsic uncertainty of a
thermodynamic state and the lack of knowledge due to coarse-graining. We
demonstrate two interpretations of observational entropy, one as the
statistical deficiency resulting from a measurement, the other as the
difficulty of inferring the input state from the measurement statistics by
quantum Bayesian retrodiction. These interpretations show that the
observational entropy implicitly includes a uniform reference prior. Since the
uniform prior cannot be used when the system is infinite-dimensional or
otherwise energy-constrained, we propose generalizations by replacing the
uniform prior with arbitrary quantum states that may not even commute with the
state of the system. We propose three candidates for this generalization,
discuss their properties, and show that one of them gives a unified expression
that relates both interpretations.
