Encoding dependence in Bayesian causal networks

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.