AI 中文总结
该研究针对分布式统计分析中密度融合的聚合顺序变异，推导了局部二阶界，证明有限三态后验对比可检测差异，所得展开在满足条件的有限树族上一致。
AI 中文摘要
分布式统计分析常通过重复成对融合来聚合局部后验或预测密度。当二元融合规则非结合时，改变有序聚合树会改变最终密度及报告的后验摘要。针对围绕共同参考密度的局部密度图中平滑f-散度平衡，我们研究这种聚合顺序变异。在平方根变换的供给权重及每次生成合并处的一阶非退化条件下，固定有限族中的每棵树共享相同的一阶系数，而首个可能依赖树的项出现在二阶。我们推导显式二阶系数，通过针对树索引二阶密度系数的递推将其传播到族中每棵树，证明有限三态后验对比可检测所得差异。该系数同时决定密度级和感兴趣量直径、旋转界、下界、标量校准的局限性及局部校正图表示，所得展开在满足所述局部条件的每个固定有限树族上一致。
英文摘要
Distributed statistical analyses often aggregate local posterior or predictive densities by repeated pairwise fusion. When the binary fusion rule is nonassociative, changing the ordered aggregation tree can change the final density and the reported posterior summaries. We study this aggregation-order variation for smooth $f$-divergence balancing in a local density chart around a common reference density. Under square-root-transformed supplied weights and first-order nondegeneracy at each generated merge, every tree in a fixed finite family shares the same first-order coefficient, whereas the first probable tree-dependent term appears at second order. We derive the explicit second-order coefficient, propagate it through every tree in the family by a recursion for the tree-indexed second-order density coefficient, and show that a finite three-state posterior contrast detects the resulting discrepancy. The same coefficient determines density-level and quantity-of-interest diameters, rotation bounds, lower bounds, a limitation of scalar calibration, and a local corrected chart representation. The resulting expansions are uniform over each fixed finite tree family satisfying the stated local conditions.
