Distributed uncertainty-management systems often combine local probabilistic models along aggregation trees chosen by communication, privacy, or scheduling constraints. The final density should depend on the weighted sources, not on the particular order in which intermediate nodes combine them. We study this requirement as an algebraic compositionality problem for binary fusion of weighted probability densities. The central question is when a local fusion rule can be executed hierarchically while remaining order-invariant. We establish a compositional boundary for local segment-valued fusion rules. Within the class of continuous binary rules with additive output weights and weight-only coefficients, order-invariant hierarchical execution characterizes normalized weighted linear pooling; norm-induced segment balancing realizes the corresponding coefficient. Smooth endpoint-to-candidate $f$-divergence balancing has a different local geometry: its quadratic expansion induces square-root effective weights, showing why pairwise solvability alone is insufficient for schedule-independent fusion. We show that this obstruction is local to endpoint-to-candidate binary balancing, whereas global divergence barycenters retain additive-weight local limits. Finally, Gaussian mixtures show how the same issue appears in finite model classes: exact fusion is compositional, whereas stepwise compression is compositional only under a congruence condition on unnormalized component measures. These results distinguish exact schedule-independent fusion from global aggregation objectives and local approximation heuristics.

翻译：分布式不确定性管理系统通常沿着由通信、隐私或调度约束选择的聚合树，组合局部概率模型。最终密度应取决于加权的信息源，而非中间节点组合它们的特定顺序。我们将这一要求研究为加权概率密度二元融合的代数组成性问题。核心问题是：何时能在保持顺序不变性的情况下，以层级方式执行局部融合规则？我们为局部段值融合规则建立了一个组成边界。在具有加性输出权重和仅权重系数的连续二元规则类别中，顺序不变层级执行刻画了归一化加权线性池化；范数诱导的段平衡实现了相应系数。光滑端点-候选$f$-散度平衡具有不同的局部几何结构：其二次展开产生平方根有效权重，这表明为何仅凭成对可解性不足以实现与调度无关的融合。我们证明这一障碍是端点-候选二元平衡特有的局部性质，而全局散度重心则保留加性权重的局部极限。最后，高斯混合模型展示了相同问题如何在有限模型类中出现：精确融合是组成性的，而逐步压缩仅当未归一化分量测度满足同余条件时才具有组成性。这些结果将精确的调度无关融合与全局聚合目标及局部近似启发式区分开来。