信息源之间既有互补也有替代，关键在是否跨越决策边界。
All Substitution Is Local
- 用贝叶斯决策框架分析信息源的交互机制
- 只有跨过决策边界时才会产生替代效应，其余地方均互补
- 适用于任意相关性的信息源，理论已用Lean 4验证
当查阅一个信息源会提升或降低另一个的价值时，何时发生？我们研究了面对有限行动的贝叶斯决策者所面临的问题。信息源间的相互作用可分解为两个相反的力量：互补力衡量一个源如何使信念移动至另一个更有效的区域，替代力衡量当前决策被解决的程度。其平衡遵循局域化原则：替代要求观测值穿越决策边界，但仅穿越边界不足以保证替代。只要后验分布仍处于当前决策区域内，替代力即消失，此时信息源必然互补，即使单个源无法单独改变决策。该结论对任意相关的源均成立，并在Lean 4中形式化。替代仅发生在决策变化的薄边界处，其余区域信息始终协作。代码与证明见：https://github.com/nidhishs/all-substitution-is-local。
原文摘要 · Abstract (English)
When does consulting one information source raise the value of another, and when does it diminish it? We study this question for Bayesian decision-makers facing finite actions. The interaction decomposes into two opposing forces: a complement force, measuring how one source moves beliefs to where the other becomes more useful, and a substitute force, measuring how much the current decision is resolved. Their balance obeys a localization principle: substitution requires an observation to cross a decision boundary, though crossing alone does not guarantee it. Whenever posteriors remain inside the current decision region, the substitute force vanishes, and sources are guaranteed to complement each other, even when one source cannot, on its own, change the decision. The results hold for arbitrarily correlated sources and are formalized in Lean 4. Substitution is confined to the thin boundaries where decisions change. Everywhere else, information cooperates. Code and proofs: https://github.com/nidhishs/all-substitution-is-local.
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