  Constraining the equation of state of cold dense matter in compact stars is a
major science goal for observing programmes being conducted using X-ray, radio,
and gravitational wave telescopes. We discuss Bayesian hierarchical inference
of parametrised dense matter equations of state. In particular we generalise
and examine two inference paradigms from the literature: (i) direct posterior
equation of state parameter estimation, conditioned on observations of a set of
rotating compact stars; and (ii) indirect parameter estimation, via
transformation of an intermediary joint posterior distribution of exterior
spacetime parameters (such as gravitational masses and coordinate equatorial
radii). We conclude that the former paradigm is not only tractable for
large-scale analyses, but is principled and flexible from a Bayesian
perspective whilst the latter paradigm is not. The thematic problem of Bayesian
prior definition emerges as the crux of the difference between these paradigms.
The second paradigm should in general only be considered as an ill-defined
approach to the problem of utilising archival posterior constraints on exterior
spacetime parameters; we advocate for an alternative approach whereby such
information is repurposed as an approximative likelihood function. We also
discuss why conditioning on a piecewise-polytropic equation of state model -
currently standard in the field of dense matter study - can easily violate
conditions required for transformation of a probability density distribution
between spaces of exterior (spacetime) and interior (source matter) parameters.
