Formalizing the presumption of independence

TL;DR

This paper explores formalizing the presumption of independence in probabilistic reasoning, proposing the concept of heuristic estimators and identifying key coherence properties, with an open problem on their existence.

Contribution

It introduces heuristic estimators as a formal approach to the presumption of independence and outlines desirable coherence properties, highlighting an open research problem.

Findings

Identifies coherence properties for heuristic estimators

Shows no existing estimators satisfy all properties

Poses an open problem on formalizing valid independence assumptions

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?