On marginals and profiled posteriors for cosmological parameter estimation

With several examples and in an analysis of the Pantheon+ supernova sample we discuss the properties of the marginal posterior distribution versus the profiled posterior distribution -- the profile likelihood in a Bayesian disguise. We investigate whether maximisation, as used for the profiling, or integration, as used for the marginalisation, is more appropriate. To report results we recommend the marginal posterior distribution.
Comment: 24 pages, 10 figures, now includes an extended motivation and discussion as well as a valuation of the Laplace approximation