  The massive black holes we observe in galaxies today are the natural
end-product of a complex evolutionary path, in which black holes seeded in
proto-galaxies at high redshift grow through cosmic history via a sequence of
mergers and accretion episodes. Electromagnetic observations probe a small
subset of the population of massive black holes (namely, those that are active
or those that are very close to us), but planned space-based gravitational-wave
observatories such as the Laser Interferometer Space Antenna (LISA) can measure
the parameters of ``electromagnetically invisible'' massive black holes out to
high redshift. In this paper we introduce a Bayesian framework to analyze the
information that can be gathered from a set of such measurements. Our goal is
to connect a set of massive black hole binary merger observations to the
underlying model of massive black hole formation. In other words, given a set
of observed massive black hole coalescences, we assess what information can be
extracted about the underlying massive black hole population model. For
concreteness we consider ten specific models of massive black hole formation,
chosen to probe four important (and largely unconstrained) aspects of the input
physics used in structure formation simulations: seed formation, metallicity
``feedback'', accretion efficiency and accretion geometry. For the first time
we allow for the possibility of ``model mixing'', by drawing the observed
population from some combination of the ``pure'' models that have been
simulated. A Bayesian analysis allows us to recover a posterior probability
distribution for the ``mixing parameters'' that characterize the fractions of
each model represented in the observed distribution. Our work shows that LISA
has enormous potential to probe the underlying physics of structure formation.
