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 lik

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.