Why not constant

Post date: Mar 22, 2021 11:44:3 PM

Scott said:

Alex asked about the solution in the ISIPTA paper to the self-sum under opposite dependence and why it isn’t a constant. I think I remember considering this issue when Ander first illustrated it, and coming to a satisfying resolution that, yes, it should be an interval. Is my pseudomnesia acting up again? I certainly can no longer remember why we might have thought that.

Ander said:

I believe the solution is correct. If you do the same operation with the equivalent p-box, you get the same interval.

Because the two credal sets are equivalent, summing under opposite dependence somehow cancels the stochasticity (or at least bounds it to an interval).

You don't get this behaviour if the fuzzy numbers aren't the same. Or at least it's not guaranteed.

I may have it wrong for op = - and C = opposite. I think you have to flip the dependence like you do in PBA. If you want to negate under opposite dependence you do levelwise with op = -.

Some MC simulations would also help the illustration. It would be nice to have a way to draw random distributions from a p-box/fuzzy number. Does anyone have any idea how to do this? I only know how to do this with a parametric p-box/fuzzy number

Scott said:

Really? Ah, right. Of course. I was thinking we were seeing discretization error. But it’s not that at all.

An “equivalent” p-box is obtained when we flip the right bound, right? So that suggests that the triangular fuzzy number [1,2,3] is equivalent to a credal set whose best p-box is env( U(1,2), U(2,3) ).

So the interval answer arises because, although we know the dependence precisely, there is ambiguity about whether we are combining the same distribution or different ones. Even if we added a stationarity assumption to the calculation of the self-sum, the answer would still not be different, because one of the distributions in the credal set is U(1,2), which would lead to the answer 3, and another distribution is U(2,3), which would lead to the answer 5. So, [3,5] is the best we can say because fuzzy numbers tend to make loose credal sets I guess.

source("C:\\Users\\ferson\\Desktop\\pba.r")

Pbox$steps = 1000

a = T(1,2,3)

oppositeconv.pbox(a,a)

Pbox$steps = 2000

a = T(1,2,3)

oppositeconv.pbox(a,a)

a = env(U(1,2),U(2,3))

convolve(a,a,opposite)

~(range=[2.99,5.01], mean=[3,5], var=[0,0.34])