Graph-based multivariate conditional autoregressive models

The conditional autoregressive model is a routinely used statistical model for areal data that arise from, for instances, epidemiological, socio-economic or ecological studies. Various multivariate conditional autoregressive models have also been extensively studied in the literature and it has been shown that extending from the univariate case to the multivariate case is not trivial. The difficulties lie in many aspects, including validity, interpretability, flexibility and computational feasibility of the model. In this paper, we approach the multivariate modelling from an element-based perspective instead of the traditional vector-based perspective. We focus on the joint adjacency structure of elements and discuss graphical structures for both the spatial and non-spatial domains. We assume that the graph for the spatial domain is generally known and fixed while the graph for the non-spatial domain can be unknown and random. We propose a very general specification for the multivariate conditional modelling and then focus on three special cases, which are linked to well-known models in the literature. Bayesian inference for parameter learning and graph learning is provided for the focused cases, and finally, an example with public health data is illustrated.