Compositional Inference for Bayesian Networks and Causality

Inference is a fundamental reasoning technique in probability theory. When applied to a large joint distribution, it involves updating with evidence (conditioning) in one or more components (variables) and computing the outcome in other components. When the joint distribution is represented by a Bayesian network, the network structure may be exploited to proceed in a compositional manner -- with great benefits. However, the main challenge is that updating involves (re)normalisation, making it an operation that interacts badly with other operations. String diagrams are becoming popular as a graphical technique for probabilistic (and quantum) reasoning. Conditioning has appeared in string diagrams, in terms of a disintegration, using bent wires and shaded (or dashed) normalisation boxes. It has become clear that such normalisation boxes do satisfy certain compositional rules. This paper takes a decisive step in this development by adding a removal rule to the formalism, for the deletion of shaded boxes. Via this removal rule one can get rid of shaded boxes and terminate an inference argument. This paper illustrates via many (graphical) examples how the resulting compositional inference technique can be used for Bayesian networks, causal reasoning and counterfactuals.