Formalizing the presumption of independence
This work addresses a foundational issue in mathematical reasoning and AI for researchers in logic, probability, and formal methods, but it is incremental as it sets up a problem without solving it.
The paper tackles the problem of formalizing the informal 'presumption of independence' heuristic used in uncertain reasoning, by introducing the concept of a heuristic estimator and proposing coherence properties, but it does not provide a solution, leaving it as an open problem.
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?