100% correct, but there are ways to push Bayesian inference back a step to justify this sort of thing.

It of course makes the problem even more complex and likely requires further approximations to computing the posterior (or even the MAP solution).

This stretches the notion that you are still doing Bayesian reasoning but can still lead to useful insights.

Probably should just call it something else then; though, I gather that the simplicity of Bayes theorom belies the complexity of what it hides.