  Hubble tension is a problem in one-dimensional (1D) posteriors, since local
$H_0$ determinations are only sensitive to a single parameter. Projected 1D
posteriors for $\Lambda$CDM cosmological parameters become more non-Gaussian
with increasing effective redshift when the model is fitted to redshift binned
data in the late Universe. We explain mathematically why this non-Gaussianity
arises and show using observational Hubble data (OHD) that Markov Chain Monte
Carlo (MCMC) marginalisation leads to 1D posteriors that fail to track the
$\chi^2$ minimum at $68\%$ confidence level in high redshift bins. To remedy
this limitation of MCMC, we resort to profile distributions as a complementary
technique. Doing so, we observe that $z \gtrsim 1$ cosmic chronometer (CC) data
currently prefers a non-evolving (constant) Hubble parameter over a
Planck-$\Lambda$CDM cosmology at $\sim 2 \sigma$. Within the Hubble tension
debate, it is imperative that subsamples of data sets with differing redshifts
yield similar $H_0$ values. In addition, we confirm that MCMC degeneracies
observed in 2D posteriors are not due to curves of constant $\chi^2$. Finally,
on the assumption that the Planck-$\Lambda$CDM cosmological model is correct,
using profile distributions we confirm a $>2 \sigma$ discrepancy with
Planck-$\Lambda$CDM in a combination of CC and baryon acoustic oscillations
(BAO) data beyond $ z \sim 1.5$. This confirms a discrepancy reported earlier
with fresh methodology.
