What frustrates me about Bayesian NNs is that talking about "priors" doesn't make nearly as much sense as it does in a regression context. A prior over parameter weights has no interpretation in the way that a prior over a regression coefficient, or even a spline smoothness, does. What you really want -- and what natural intelligence probably has -- are priors over aspects of the world.

Francois Chollet's paper on measuring intelligence was really informative for me on this front; the "priors" you should have about the world are not half-cauchys over certain hyperparameters or whatever, but priors about agent-ness, object-ness, goal-oriented-ness, and so on. How to encode that in a network...well, that's the real trick, right?

I agree that priors over aspects of the world would be more useful, but I don't think that they're important in making natural intelligence powerful. In my experience, the important thing is to make your prior really broad, but containing all kinds of different hypotheses with different kinds of rich structure.

I claim that knowing a priori about things like agents and objects just doesn't save you all that much data, as long as you have the imagination to consider all structures at least that complex.