Dempster-Shafer and modelling beliefs about sets (emiruz.com)

A user-facing problem: you have a set X of binary variables and receive evidence about arbitrary subsets of X expressed as quantified logic formulas (e.g., “some pair in A disagree” or “all members of Q are 1”). These statements range over concrete assignments to X, so they’re not about atomic events and there are no hard facts to condition on — making a straightforward Bayesian update awkward. The note shows that Dempster–Shafer (DS) theory naturally handles this by placing mass functions over the powerset of the space of all assignments (m: 2^{bar X} → [0,1]), combining independent information sources with Dempster’s rule (m_{1,2}(A) ∝ sum_{B∩C=A} m1(B)m2(C) with normalization 1−K where K measures conflict), and deriving belief bel(A)=∑_{B⊆A} m(B) and plausibility pl(A)=∑_{B: A∩B≠∅} m(B) as lower/upper bounds on probabilities implied by the logic-constraint evidence. The author prototypes this in GNU SETL on five binary vars with four logical updates (each gives mass to a constraint and leftover vacuous mass), fuses them, and queries three events: beliefs come out 0.8, 0.9, 0.98 with plausibilities all 1. The takeaway: DS offers an elegant, prior-free way to aggregate overlapping, non-atomic logical evidence (useful for sensor fusion and constraint-based inference) but has pitfalls — Dempster’s rule lacks Bayesian optimality guarantees, can yield counterintuitive results, and is computationally heavy since masses live on 2^{assignments}. A frequentist alternative via constrained simulation is possible but less elegant and potentially intractable.