Dempster-shafer and reasoning about sets
emiruz.comDS theory has never been popular. Even Bayesianism (cf. "Bayesian nonparametrics" two decades ago) took a back seat after LLMs came out. DS theory, with its greater computational complexity, is therefore even less attractive.
Yes and we are IMHO worse off, without a proper theory of reasoning under uncertainty, it's difficult to understand what LLMs are doing (or even hard to define what they should be doing).
I agree that DS is computationally prohibitive, but another way out (aside from probability, which I don't like either) is with various systems of fuzzy logic (or you can just go with the most expressive one under the lovely name \L\Pi 1/2).
(BTW I am also exploring approach to uncertainty based on untyped lambda calculus, where each term is interpreted as a kind of "model of the world". Uncertainty degree is given by whether the term has a normal form or not. If it has not, then it is certain, while if it has a normal form, it means that additional assumptions/arguments need to be supplied to specify the model further.)
I think that's overly reductivist. In the general case DS operates on up to 2^M sets where M is the cardinality of the hypothesis space: worst case scenario. That's not true if hypotheses are hierarchical, or if evidence is frequently about the same set, or there just isn't enough evidence to fuse to get to 2^M.
In the worst case scenario there are efficient approximation methods which can be used.
This reminds me of `PrSAT`, a satisfier for probabilistic statements. ("Does a distribution exist that satisfies the following constraints?").
See: https://fitelson.org/PrSAT/, and the linked paper: https://fitelson.org/pm.pdf
The paper starts off slow, but have patience to read up to section 4, Applications, which is kind of surprising.
Thanks for this reference; I found this paper interesting, but it is a satisfiability solver. Inherently it cannot quantify the probability of a subset of events, but it can find a probability assignment given a set of constraints. I.e. prove possibility. More usefully it can show that no such assignment is possible.
I don't understand what the notation means.
For example when the author says:
P(Q ⊆ X | ∀ x ∈ Q (x = 1))
This is equivalent to P(Q ⊆ X | Q = {1}), which further simplifies to P(1∈X).
This seems to be a type error (isn't X supposed to be a set of binary variables?), and also an awfully cumbersome way to write P(1∈X).
Anyone have some idea what the article is trying to say?
It's a typo. Its supposed to be a comma not a pipe, and read P(Q ⊆ X , ∀ x ∈ Q (x = 1)). I.e. Q is some subset of X and for all x in Q, x=1.
What a dempster fire