Bayesian (belief, learning, or causal) networks (BNs) represent complex, uncertain spatio-temporal dynamics by propagation of conditional probabilities between identifiable "states" with a testable causal interaction model. Typically, they assume random variables are discrete in time and space, with a static network structure that may evolve over time, according to a prescribed set of changes over a successive set of discrete model time-slices (i.e., snap-shots). But the observations that are analyzed are not necessarily independent and are autocorrelated due to their locational positions in space and time. Such BN models are not truly spatial-temporal, as they do not allow for autocorrelation in the prediction of the dynamics of a sequence of data. We begin by discussing Bayesian causal networks and explore how such data dependencies could be embedded into BN models from the perspective of fundamental assumptions governing space-time dynamics. We show how the joint probability distribution for BNs can be decomposed into partition functions with spatial dependence encoded, analogous to Markov Random Fields (MRFs). In this way, the strength and direction of spatial dependence both locally and non-locally could be validated against cross-scale monitoring data, while enabling BNs to better unravel the complex dependencies between large numbers of covariates, increasing their usefulness in environmental risk prediction and decision analysis.Keywords: Bayesian, causal, clique, dependence, GIS, network
ENVIRONMENTAL INFORMATICSA wide array of statistical modeling approaches are available for exploring real-world complex spatio-temporal phenomena including multivariate regression, multivariate analysis of variance (MANOVA), principal components analysis (PCA), canonical correlation analysis (CCA), factor analysis, spectral analysis, vector autoregressive models (VARs), and machine-learning tree and expert rule-based techniques-to name just a few (Izenman, 2013). Deciding which technique to employ relies heavily on the main purpose of a model in a given application context (e.g., estimation, prediction, and problem dimensional-reduction), the availability and quality of data and auxiliary information, and the level of accuracy required to aid in decision-making. Given the large uncertainties facing environmental decision making, and the need for more robust prediction methods, Bayesian-hierarchical, causal-learning, and copula-based network techniques are increasingly being employed to better resolve interactions and trade-offs in observed, latent, and decision-making variables. These models also provide a tractable way to integrate both tradition and more modern forms of data-from station monitoring networks, sensor-networks, and remote-sensing/satellite imagery, and can prove computationally challenging to solve numerically and to validate in the case of high-dimensional model systems.