Spatial clustering increasingly involves complex objects, including probability distributions, networks, and structured matrices, for which conventional parametric likelihoods may be difficult to specify. We construct a generalized likelihood by exponentiating a cluster-specific Bregman loss on fixed representations of the responses, avoiding the need to specify a parametric sampling model for each response class. Each cluster is associated with an unknown representative, and the generalized likelihood measures within-cluster homogeneity through the Bregman divergence between the represented observations and that representative. A prior family matched to the Bregman geometry yields conjugate generalized posterior updates and permits exact marginalization of the cluster-specific representatives. The resulting collapsed scores combine within-cluster Bregman dispersion with uncertainty in the representatives, while a spanning-tree partition prior restricts posterior partition support to spatially contiguous clusters. This collapsed representation also leads to a tractable MCMC algorithm for posterior computation. The framework accommodates multiple complex-object response classes within a common inferential construction. We establish posterior consistency for the spatial partition and cluster-specific representatives under infill-domain asymptotics. Simulation studies with different complex-object responses demonstrate accurate recovery of spatial partitions and cluster-specific representatives. We further analyze racial-composition distributions across spatial units in Houston, illustrating geographically coherent clusters with distinct demographic compositions.

翻译：暂无翻译