  We perform the first convergent Bayesian global fits of 4D Composite Higgs
Models with partially-composite third generation quarks and leptons based on
the minimal $SO(5) \rightarrow SO(4)$ symmetry breaking pattern. We consider
two models with the $\tau$ lepton and its associated neutrino in different
representations of $SO(5)$. Fitting each model with a wide array of
experimental constraints allows us to analyse the Bayesian evidence and
currently-observed fine-tuning of each model by calculating the
Kullback-Leibler divergence between their respective priors and posteriors.
Both models are found to be capable of satisfying all constraints at the
$3\sigma$ level, although the model with leptons in the $\mathbf{14}$ and
$\mathbf{10}$ representations is strongly preferred over those in the
$\mathbf{5}$ representation from a Bayesian viewpoint. Finally, we consider the
experimental signatures for the preferred parameters in these models, including
lepton partner decay signatures and gluon-fusion produced Higgs signal
strengths, and discuss their potential phenomenology at future high-luminosity
LHC runs.

High Energy Physics - Phenomenology