  High-$\langle \beta \rangle$ operations require a fast and robust inference
of plasma parameters with a self-consistent MHD equilibrium. Precalculated MHD
equilibria are usually employed at W7-X due to the high computational cost. To
address this, we couple a physics-regularized NN model that approximates the
ideal-MHD equilibrium with the Bayesian modeling framework Minerva. We show the
fast and robust inference of plasma profiles (electron temperature and density)
with a self-consistent MHD equilibrium approximated by the NN model. We
investigate the robustness of the inference across diverse synthetic W7-X
plasma scenarios. The inferred plasma parameters and their uncertainties are
compatible with the parameters inferred using the VMEC, and the inference time
is reduced by more than two orders of magnitude. This work suggests that MHD
self-consistent inferences of plasma parameters can be performed between shots.
