双边界条件推断：双边界乘积规则及其意义
Dual Boundary Condition Inference: A Two-Boundary Product Rule and Its Implications
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中文总结 AI 辅助
本文研究双边界条件推断（DBCI），通过乘积与重归一化结合前向和后向约束，分类其指数选择，提出双边界可约性准则，并揭示与贝叶斯规则及ABL规则的联系。
中文摘要 AI 辅助
许多推断问题受到来自两方面的约束：前向信息和对结果的第二个约束。双边界条件推断（DBCI）将两者相乘并重新归一化。该规则并非新规则：它是指数乘积形式，并且是对数池化族中的单位指数成员。本文贡献在于分类和可约性刻画。代数并未固定输入组合时的指数：给定统计量 log p_fwd + log p_bwd（该统计量已包含乘积形式），最大熵提供了族 (p_fwd p_bwd)^lambda 并选择不出任何成员。如果将输入视为两个等权重的意见，则三个要求选择 lambda = 1/2：一致同意保持、在共同贝叶斯更新下的一致性以及最小对称 Kullback-Leibler 散度。如果将它们视为分别应用的因子，则两个要求选择 lambda = 1：中性的第二输入必须使第一输入保持不变，并且对一个输入的更新必须原样传递。对于秩一中间投影测量，Aharonov-Bergmann-Lebowitz（ABL）规则实现了该形式且无额外参数。每个正指数对结果的排序相同，因此该选择对于选择最可能的结果是不可见的，但对于按总质量对类别进行评分则不是。在相关的识别下，相同的乘积形式给出贝叶斯规则，而正比于恒等式的最终效应返回普通的正向 Born 概率。第二个结果是双边界可约性准则：DBCI 恰好在前向边界上可约，当且仅当有效后向边界（在正重缩放意义下）可约。因此，固定的先验或模型固定的结构约束不会在共享相同前向边界的案例之间增加区分，尽管它可能编码大量信息并仍然实质性地影响结果。
英文摘要
Many inference problems are constrained from two sides: forward information and a second constraint on outcomes. Dual Boundary Condition Inference (DBCI) multiplies them and renormalises. The rule is not new: it is the product-of-experts form and the unit-exponent member of the logarithmic-pooling family. The paper contributes a classification and reducibility characterisation. The algebra does not fix the exponent at which the inputs combine: given the statistic log p_fwd + log p_bwd, which already builds in the product form, maximum entropy supplies the family (p_fwd p_bwd)^lambda and selects no member. If the inputs are read as two equally weighted opinions, three requirements select lambda = 1/2: unanimity preservation, consistency under a common Bayesian update, and minimal symmetric Kullback-Leibler divergence. If they are read as separately applied factors, two requirements select lambda = 1: a neutral second input must leave the first unchanged, and an update to one input must pass through unchanged. For a rank-one intermediate projective measurement, the Aharonov-Bergmann-Lebowitz (ABL) rule realises this form with no extra parameter. Every positive exponent orders outcomes identically, so the choice is invisible to picking the likeliest outcome, but not to scoring a class by total mass. Under the relevant identifications, the same product form gives Bayes' rule, while a final effect proportional to the identity returns the ordinary forward Born probabilities. The second result is a Two-Boundary Reducibility Criterion: DBCI factors through the forward boundary exactly when the effective backward boundary, up to positive rescaling, does. So a fixed prior or model-fixed structural constraint adds no distinctions between cases sharing the same forward boundary, though it may encode substantial information and still materially affect the result.
发表机构
- European Institute of Science in Management (EISM)（欧洲科学管理研究所）
- Faculty of Engineering and the Institute of Nanotechnology and Advanced Materials, Bar-Ilan University（巴伊兰大学工程与纳米技术及先进材料研究所）
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