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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.

At some level, you have to choose something. You can't know every level in your hierarchy.

Sure, instead of saying "choose" a prior, you could say "elicit". But I think in this context, focusing on a practitioner's prior knowledge is missing the point. For the sorts of problems we use NNs for, we don't usually think that the guy designing the net has important knowledge that would help making good predictions. Choosing a prior is just an engineering challenge, where one has to avoid accidentally precluding plausible hypotheses.