Formalizing the presumption of independence
Abstract
Mathematical proof aims to deliver confident conclusions, but a very similar process of deduction can be used to make uncertain estimates that are open to revision. A key ingredient in such reasoning is the use of a "default" estimate of $\mathbb{E}[XY] = \mathbb{E}[X] \mathbb{E}[Y]$ in the absence of any specific information about the correlation between $X$ and $Y$, which we call *the presumption of independence*. Reasoning based on this heuristic is commonplace, intuitively compelling, and often quite successful  but completely informal. In this paper we introduce the concept of a heuristic estimator as a potential formalization of this type of defeasible reasoning. We introduce a set of intuitively desirable coherence properties for heuristic estimators that are not satisfied by any existing candidates. Then we present our main open problem: is there a heuristic estimator that formalizes intuitively valid applications of the presumption of independence without also accepting spurious arguments?
 Publication:

arXiv eprints
 Pub Date:
 November 2022
 DOI:
 10.48550/arXiv.2211.06738
 arXiv:
 arXiv:2211.06738
 Bibcode:
 2022arXiv221106738C
 Keywords:

 Computer Science  Artificial Intelligence;
 Computer Science  Discrete Mathematics;
 Computer Science  Logic in Computer Science
 EPrint:
 61 pages, 8 figures