Post date: Mar 22, 2021 6:43:49 PM
Ander's presentation at ISIPTA 2021 (paper, poster, and slides attached at the bottom of this page) was well received and garnered a lot of attention.
Ander said:
I think we got a good response in ISIPTA, I'm very pleased.
I love how Sebastien was so on the ball about the random set dependence. I literally showed up at the poster and he was asking about it. [The] back and forth with him was great.
Scott said:
Sébastien said, and of course I concede, that he shouldn’t have to give a counterexample, but he also admitted that he knows no particular reason to think that the elementary arithmetic operations would violate the conjecture.
We are confident that so long as the conjecture works for each of the elementary operations, that applying these operations sequentially will also work, so long as the function is decomposable into a sequence of elementary operations, aren’t we? Sébastien worries about strange functions that look into themselves. Maybe they’re not as strange as I’d like to think, but that’s not the question at hand.
Schmelzer should probably worry us more immediately. After we debrief on Wednesday, we can arrange a time to chat with him.
I should admit that as far as I know there is no proof that p-box arithmetic under independence is valid. I certainly never produced one. Williamson based his work on the Frank-Nelsen-Sklar and Makarov proofs about Fréchet convolutions, but I’m pretty sure he didn’t offer a proof about independence, and I don’t think Yager did either. Like every Sylvester ‘theorem’, I believe it as much as I believe anything in mathematics. But I remember that this belief came from numerical experiments, not a proof. It’s not the sort of thing that proofs really convince one about, which are really just for talking to others.
In my old age, I’m not sure I can still even state such a theorem conjecture. Maybe we need that flaky text about structured bounding spaces. Noodling this afternoon reminds me of the Russian expression “trying to stand on your ears”.
Every Dempster-Shafter structure defines a credal set. Let C(d) denote the credal set defined by a Dempster-Shafter structure d.
Let X denote a credal set of distributions on the real line, and let D(X) be any Dempster-Shafer structure on the real line such that C(D(X)) encloses X, i.e., such that every element of X is also an element of C(D(X)) defined by any Dempster-Shafer structure of X. Identifying the best-possible such Dempster-Shafer structure is not always possible, and the credal set defined by even the best-possible Dempster-Shafer structure may not be the original X.
For credal sets on the real line X and Y, use X+Y to denote the credal set composed of sigma convolutions F + G, where F Î X and G Î Y (and + denotes sigma convolution which assumes “strong” stochastic independence). X+Y is a subset of the credal set defined by the Dempster-Shafer structure obtained as the Cartesian product of (any set of) condensed slivers of Dempster-Shafer structures D(X) and D(Y) assuming random-set independence. That is,
X+Y Í C( D(X) + D(Y) )
where + denotes Yager’s Cartesian product on Dempster-Shafer structures with random-set independence. Can this be expressed more simply?
P-boxes are also a structured bounding space for the set of distributions on the real line. Every p-box b thus defines a credal set C(b) consisting of all the distributions that stochastically dominate the left edge of the p-box, and which are stochastically dominated by the right edge of the p-box.
For any credal set X, let B(X) denote any p-box such that X Í C(B(X)). Because p-boxes are crude, X may be smaller than C(B(X)). So then we just want to prove that
X + Y Í C( B(X) + B(Y) ).
I presume a proof would let B(X) = [L(X), R(X)] denote any of several possible p-boxes of X, where L(X) and R(X) are the left and right edge distributions, such that every element of X stochastically dominates L(X) and is stochastically dominated by R(X).
C( [L(X), R(X)] ) likewise denotes the credal set implied by the p-box [L(X), R(X)] of X if it exists. [If it doesn’t exist, I guess we’d want to define the space of distributions on the extended reals so we can have infinities.]
Suppose F Î X and G Î Y. Their independent convolution F+G is an element of C( B(X) + B(Y) ) if it is an element of
C( [L(X), R(X)] + [L(Y), R(Y)] )
= C( [L(X) + L(Y), R(X) + R(Y)] )
= { H : (R(X) + R(Y))(z) £ H(z) £ (L(X) + L(Y))(z), for all real values z }
which will be so if (R(X) + R(Y))(z) £ (F+G)(z) £ (L(X) + L(Y))(z) for all z.
Obviously, the truth of this depends on the simplicity of the additive convolution operation +. Addition is 2-increasing, but do I know how to show that independent sigma convolution is 2-increasing with respect to the partial ordering of stochastic dominance?
Of course, p-boxes are cruder than Dempster-Shafer structures, so C(d) may be smaller than C(B(d)).
Maybe it’s silly to look for a simple set & operator notation. I tried to use the notation in your ISIPTA paper, but I was getting lost in the p with seemingly complex superscripts and subscripts. Why does it even have the X subscript in expression (4)? Maybe, like Nec and Bel, we should at least use Left and Right for the edges of a p-box?
I’m not sure what objects we should consider primary in this discussion. Maybe it should be formulated by using something different:
Credal set P-box Dempster-Shafer structure
X B(X) D(X)
C(b) b D(b)
C(d) B(d) d
Or maybe it is preferable to stick on the first line because otherwise it is cumbersome to express the fact that X Í C(D(X)) Í C(B(X)). Ah, maybe we should make the (uncertain) random variable the primary object as you do in the ISIPTA paper. Or maybe its probability measure or distribution. Then C(X) is the credal set representing the uncertainty about a random variable X?
I’m using the words ‘defines’ and ‘defined by’ as in the ISIPTA paper to talk about the credal set associated with a p-box or Dempster-Shafer structure, but we might prefer ‘implies’ and ‘implied by’.