Recently, Bayesian probability theory has been used at a number of experiments to fold uncertainties and interdependencies in the diagnostic data and forward models, together with prior knowledge of the state of the plasma, to increase accuracy of inferred physics variables. A new probabilistic framework, MINERVA, based on Bayesian graphical models, has been used at JET and W7-AS to yield predictions of internal magnetic structure. A feature of the framework is the Bayesian inversion for poloidal magnetic flux without the need for an explicit equilibrium assumption. Building on this, we discuss results from a new project to develop Bayesian inversion tools that aim to 1 distinguish between competing equilibrium theories, which capture different physics, using the MAST spherical tokamak, and 2 test the predictions of MHD theory, particularly mode structure, using the H-1 Heliac. Specifically, we report on correction of the motional Stark effect, pickup coils, flux-loop constrained Bayesian inferred equilibrium for varying toroidal flux.