FWIW, it's not really clear to me how to map the IR/higher-language analogy into this situation, but when I try to it feels more like belief functionals are the higher-level language and PDGs are a particular lower-level implementation thereof (and from this perspective it doubly makes sense why the map isn't injective, since there are a wide variety of ways to compile a given higher-level program into a given lower-level representation); I don't really see how it could be the case that PDGs are less burdensome than belief functionals given that if you wan...
I'd like to add another point here regarding burden. The fact that davidad didn't make this point is surprising to me, so I may be misunderstanding something here - please let me know if so!
I want to push back on your point that "the burden of selecting a lower semi-continuous function on distributions (over which variables?) is WAY higher, since there are far more choices to make, and less standard guidance on how to make them" - to do so I will draw an analogy. Consider a problem which asks the user for a function satisfying some properties on some space...
This post is absolutely phenomenal and I am genuinely baffled that there's not an abundance of comments here. I'll be sharing this with anyone who'll listen...