Bayesian Contiguous Gaussian and Generalized Linear Dynamic Models on Graphs: Theory, Computation, and Simulation

We study graph-indexed time series in which the vertices of a connected graph are partitioned into spatially contiguous clusters and each cluster carries a Gaussian or generalized linear dynamic state-space regression. The observation layer covers continuous Gaussian responses and binomial, Poisson, and negative-binomial GLM responses, while the clusterlevel state is a time-varying regression vector shared by all vertices in the same connected region. The partition prior is represented through cuts of spanning trees, which enforces connected clusters while allowing irregular shapes. We establish support properties of the prior, exact-target invariance for reversible-jump and tempered kernels, posterior concentration for dynamic predictors under non-i.i.d. Gaussian/GLM likelihoods, and partition-selection consistency under a separation condition. We then develop a posterior sampler based on split, merge,cut-swap, tree-refresh, boundary-reassignment, state-hyperparameter, and observation-parameter moves, together with proxy-guided proposals, locally balanced options, and standard parallel tempering. We also describe pointwise and simultaneous functional credible bands for dynamic coefficients. A controlled negative-binomial simulation illustrates the data-generating mechanism, recovery of the connected partition, the role of parallel tempering in improving exploration, and posterior estimation of main-effect and interaction trajectories.