Possibility to infer $q$ probabilistically as well as $\phi$?

[SamDuffield]

Would it be possible to do Bayesian inference over $(\phi, q)$ rather than just $\phi$?

Potentially accepting some prior over $q$?

This could easily be extended in the case of sequential Monte Carlo Bayesian inference, I wonder how it could be done in terms of exact (Fourier) or moment-matching (von Mises) - perhaps could make an independence approximation $p(\phi, q) = p(\phi) p(q)$.

The nice thing is that this could be done without additional quantum overhead (i.e. same number of shots per update).