I don’t think this works (or at least I don’t understand how). What space are you even mapping here (I think your xi are samples, so {0,1}N to itself?) and what’s the operation on those spaces, and how does that imply the kind of symmetry from the OP?
There’s the automorphism
(x1,x2,x3,…,xn)↦(x1,x1⊕x2,x2⊕x3,…,xn−1⊕xn)
which turns a switchy distribution into a sticky one, and vice versa. The two have to be symmetric, so your conclusion cannot be correct.
(1,1,1,1,1,1,1,1,1) maps to (1,0,0,0,0,0,0,0,0).
You probably meant prefix sums instead of pairwise sums.
In any case, Bayesian reasoning is not symmetrical with respect to any given automorphism, unless the hypotheses space is.
Then all zeroes maps to all zeroes.
I don’t think this works (or at least I don’t understand how). What space are you even mapping here (I think your xi are samples, so {0,1}N to itself?) and what’s the operation on those spaces, and how does that imply the kind of symmetry from the OP?