Estimates of the Hubble constant, $H_0$, from the local distance ladder and from the cosmic microwave background (CMB) are discrepant at the ~3σ level, indicating a potential issue with the standard Λ cold dark matter (ΛCDM) cosmology. A probabilistic (i.e. Bayesian) interpretation of this tension requires a model comparison calculation, which in turn depends strongly on the tails of the $H_0$ likelihoods. Evaluating the tails of the local $H_0$ likelihood requires the use of non-Gaussian distributions to faithfully represent anchor likelihoods and outliers, and simultaneous fitting of the complete distance-ladder data set to ensure correct uncertainty propagation. We have hence developed a Bayesian hierarchical model of the full distance ladder that does not rely on Gaussian distributions and allows outliers to be modelled without arbitrary data cuts. Marginalizing over the full ~3000-parameter joint posterior distribution, we find $H_0$ = (72.72 ± 1.67) km s-1 Mpc-1 when applied to the outlier-cleaned Riess et al. data, and (73.15 ± 1.78) km s-1 Mpc-1 with supernova outliers reintroduced (the pre-cut Cepheid data set is not available). Using our precise evaluation of the tails of the $H_0$ likelihood, we apply Bayesian model comparison to assess the evidence for deviation from ΛCDM given the distance-ladder and CMB data. The odds against ΛCDM are at worst ~10:1 when considering the Planck 2015 XIII data, regardless of outlier treatment, considerably less dramatic than naïvely implied by the 2.8σ discrepancy. These odds become ~60:1 when an approximation to the more-discrepant Planck Intermediate XLVI likelihood is included.