Compositional Boundaries for Density Fusion
For researchers in distributed uncertainty management and probabilistic fusion, the paper provides a theoretical boundary for when hierarchical fusion can be order-invariant, distinguishing exact fusion from approximations.
The paper studies when hierarchical fusion of weighted probability densities is order-invariant, showing that normalized weighted linear pooling is uniquely characterized by this property among continuous binary rules with additive output weights. It identifies that endpoint-to-candidate f-divergence balancing fails schedule-independence due to square-root effective weights, while global divergence barycenters retain additive-weight limits.
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.