Recursive Causality in Bayesian Networks and Self-Fibring Networks

So causal models need to be able to treat causal relationships as causes and effects. This observation motivates an extension the Bayesian network causal calculus (Section 2) to allow nodes that themselves take Bayesian networks as values. Such networks will be called recursive Bayesian networks (Section 3). Because recursive Bayesian networks make causal and probabilistic claims at different levels of their recursive structure, there is a danger that the network might contradict itself. Hence we need to ensure that the network is consistent, as explained in Section 4. Having done this, in Section 5 we propose a new Markov condition: under this condition a recursive Bayesian network determines a joint probability distribution over its domain. In Section 6 we compare our approach to other generalisations of Bayesian networks, and in Section 7 we show by analogy with recursive Bayesian networks how recursive causality can be modelled in structural equation models. A similar analogy motivates the application of recursive Bayesian networks to a non-causal domain, namely the modelling of arguments (Section 8). A recursive Bayesian network is an instance of a very general structure called a self-fibring information network, whose properties are explored in Section 9 and Section 10.