  Monte Carlo (MC) generators are crucial for analyzing data in particle
collider experiments. However, often even a small mismatch between the MC
simulations and the measurements can undermine the interpretation of the
results. This is particularly important in the context of LHC searches for rare
physics processes within and beyond the standard model (SM). One of the
ultimate rare processes in the SM currently being explored at the LHC, $pp\to
t\bar tt \bar t$ with its large multi-dimensional phase-space is an ideal
testing ground to explore new ways to reduce the impact of potential MC
mismodelling on experimental results. We propose a novel statistical method
capable of disentangling the 4-top signal from the dominant backgrounds in the
same-sign dilepton channel, while simultaneously correcting for possible MC
imperfections in modelling of the most relevant discriminating observables --
the jet multiplicity distributions. A Bayesian mixture of multinomials is used
to model the light-jet and $b$-jet multiplicities under the assumption of their
conditional independence. The signal and background distributions generated
from a deliberately mistuned MC simulator are used as model priors. The
posterior distributions, as well as the signal and background fractions, are
then learned from the data using Bayesian inference. We demonstrate that our
method can mitigate the effects of large MC mismodellings in the context of a
realistic $t\bar tt\bar t$ search, leading to corrected posterior distributions
that better approximate the underlying truth-level spectra.
