There is no such thing as burden of proof outside of certain structured settings.
I think this is important. There isn't much epistemological high ground to be claimed by anyone. If I claim that the other side bears a burden of proof, I'm usually just admitting that I can't prove my case any more than they can prove theirs, but I want assertion without proof to be considered bad when they do it, and not when I do it. Asserting "not-A" is just as much an assertion as asserting "A". Whether it's zero or one, every bit is a bit.
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physics Guy observation is both simple and sharp. It may not discourage the use of the phrase burden of proof though discussion of different views would be aided by absence of the phrase.