On the use of the local prior on the absolute magnitude of Type Ia
  supernovae in cosmological inference
  A dark-energy which behaves as the cosmological constant until a sudden
phantom transition at very-low redshift ($z<0.1$) seems to solve the >4$\sigma$
disagreement between the local and high-redshift determinations of the Hubble
constant, while maintaining the phenomenological success of the $\Lambda$CDM
model with respect to the other observables. Here, we show that such a
hockey-stick dark energy cannot solve the $H_0$ crisis. The basic reason is
that the supernova absolute magnitude $M_B$ that is used to derive the local
$H_0$ constraint is not compatible with the $M_B$ that is necessary to fit
supernova, BAO and CMB data, and this disagreement is not solved by a sudden
phantom transition at very-low redshift. We make use of this example to show
why it is preferable to adopt in the statistical analyses the prior on $M_B$ as
an alternative to the prior on $H_0$. The three reasons are: i) one avoids
potential double counting of low-redshift supernovae, ii) one avoids assuming
the validity of cosmography, in particular fixing the deceleration parameter to
the standard model value $q_0=-0.55$, iii) one includes in the analysis the
fact that $M_B$ is constrained by local calibration, an information which would
otherwise be neglected in the analysis, biasing both model selection and
parameter constraints. We provide the priors on $M_B$ relative to the recent
Pantheon and DES-SN3YR supernova catalogs. We also provide a Gaussian joint
prior on $H_0$ and $q_0$ that generalizes the prior on $H_0$ by SH0ES.
