Bayesian Inference to infer causal structures

Time varying processes in nature are often complex with non-linear and non-gaussian components. Complexity of environments and processes make it hard to disentangle different causal mechanisms which drive the observed time-series, and to make forecasts. There is a revolution in causal discovery methods for the geosciences, from simple to quite comprehensive, and recently even arguably complete. However, it is often unclear what the inferred causal strengths mean. Uncertainty quantification in causal discovery is still in its infancy, mainly focussing on wether there is a causal relation or not using significance testing. Here we propose a full Bayesian treatment of causal structures, enabling full uncertainty quantification and to update a causal structure in a principled way when new information becomes available. This new information can either be in terms of longer time series, but also the inclusion of new hitherto unknown causal processes.

In this presentation the basic Bayesian methodology is explained and applied to a new causal discovery framework. This framework can unravel nonlinear interactions among driver processes and is complete in that it determines the size of the missing processes. Application to simple systems like Lorenz 1963 show that it allows for both inferring the structure of the underlying differential equations and the large-scale dynamics of the underlying attractor. Interestingly, the resulting causal structure cannot be represented by standard graphical models, so Bayesian network theory cannot be applied. We will discuss how this causal framework is embedded in the Bayesian framework, including how to think about the inference process in this case, so what is the prior, the likelihood, and the observation operator. The usefulness of the theory will be demonstrated by applying it to toy models and simple systems.