  The crust-core phase transition of neutron stars is quantitatively studied
within a unified meta-modelling of the nuclear Equation of State (EoS). The
variational equations in the crust are solved within a Compressible Liquid Drop
(CLD) approach, with surface parameters consistently optimized for each EoS set
on experimental nuclear mass data. When EoS parameters are taken from known
Skyrme or RMF functionals, the transition point of those models is nicely
reproduced. A model-independent probability distribution of EoS parameters and
of the transition density and pressure is determined with a Bayesian analysis,
where the prior is given by an uncorrelated distribution of parameters within
the present empirical uncertainties, and constraints are applied both from
neutron star physics and ab-initio modelling. We show that the characteristics
of the transition point are largely independent of the high density properties
of the EoS, while ab-initio EoS calculations of neutron and symmetric matter
are far more constraining. The most influential parameter for the determination
of the transition point governs the surface properties of extremely neutron
rich matter, and it is strongly unconstrained. This explains the large
dispersion of existing predictions of the transition point. Only if the surface
tension is fixed to a reasonable but somewhat arbitrary value, strong
correlations with isovector parameters ($L_{sym},K_{sym}$ and $Q_{sym}$) are
recovered. Within the present experimental and theoretical uncertainties on
those parameters, we estimate the transition density as $n_t= 0.072\pm 0.011$
fm$^{-3}$ and the transition pressure as $P_t=0.339\pm0.115$ MeV fm$^{-3}$.
