  Heavy-ion collisions provide a window into the properties of many-body
systems of deconfined quarks and gluons. Understanding the collective
properties of quarks and gluons is possible by comparing models of heavy-ion
collisions to measurements of the distribution of particles produced at the end
of the collisions. These model-to-data comparisons are extremely challenging,
however, because of the complexity of the models, the large amount of
experimental data, and their uncertainties. Bayesian inference provides a
rigorous statistical framework to constrain the properties of nuclear matter by
systematically comparing models and measurements.
  This review covers model emulation and Bayesian methods as applied to
model-to-data comparisons in heavy-ion collisions. Replacing the model outputs
(observables) with Gaussian process emulators is key to the Bayesian approach
currently used in the field, and both current uses of emulators and related
recent developments are reviewed. The general principles of Bayesian inference
are then discussed along with other Bayesian methods, followed by a systematic
comparison of seven recent Bayesian analyses that studied quark-gluon plasma
properties, such as the shear and bulk viscosities. The latter comparison is
used to illustrate sources of differences in analyses, and what it can teach us
for future studies.
