Formalizing the presumption of independence
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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 𝔼[XY] = 𝔼[X] 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?
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