  Monte Carlo (MC) algorithms are commonly employed to explore high-dimensional
parameter spaces constrained by data. All the statistical information obtained
in the output of these analyses is contained in the Markov chains, which one
needs to process and interpret. The marginalization technique allows us to
digest these chains and compute the posterior distributions for the parameter
subsets of interest. In particular, it lets us draw confidence regions in
two-dimensional planes, and get the constraints for the individual parameters.
It is very well known, though, that the marginalized results can suffer from
volume effects, which can introduce a non-negligible bias into our conclusions.
The impact of these effects are barely studied in the literature. In this paper
we first illustrate the problem through a very clear and simple example in two
dimensions, and suggest the use of the profile distributions (PDs) as a
complementary tool to detect marginalization biases directly from the MC
chains. We apply our method to four cosmological models: the standard
$\Lambda$CDM, early dark energy, coupled dark energy and the Brans-Dicke model
with a cosmological constant. We discuss the impact of the volume effects on
each model and the cosmological tensions, using the full Planck 2018
likelihood, the Pantheon compilation of supernovae of Type Ia and data on
baryon acoustic oscillations. Our test is very efficient and can be easily
applied to any MC study. It allows us to estimate the PDs at a derisory
computational cost not only for the main cosmological parameters, but also for
the nuisance and derived ones, and to assess the need to perform a more
in-depth analysis with the exact computation of the PDs.
