Influence diagrams for causal modelling and inference

We consider a variety of ways in which probabilistic and causal models can be represented in graphical form. By adding nodes to our graphs to represent parameters, decisions, etc., we obtain a generalisation of influence diagrams that supports meaningful causal modelling and inference, and only requires concepts and methods that are already standard in the purely probabilistic case. We relate our representations to others, particularly functional models, and present arguments and examples in favour of their superiority.