Definition: Operational influence

Fix an operational process and an accessible output at $q$. Two admissible interventions $\mathcal{I},\mathcal{J}$ at $p$ must have common comparison interfaces; all other operations and preparations are held fixed. Let ${\Phi }_{q\mid \mathcal{I}}$ and ${\Phi }_{q\mid \mathcal{J}}$ be the resulting complete output channels. Set

Write $p{R}_{\inf }q$ when ${\Delta }_{p\to q}>0$.

For an interface containing only classical records, the corresponding definition is a change in the unconditional output marginal under the intervention. In a process-matrix formulation the admissible instrument insertions and signalling comparison must be specified within that formulation.