lin-0075

5.3 Smooth intervention protocols

Pearl’s interventions and potential-outcome estimands are usually formulated at finite scale [ Pearl , 2009b , Rubin , 2005 ] . To differentiate an intervention, we need a smooth path that connects it to the observational model.

Definition 5.3 Smooth intervention protocol

A smooth local intervention protocol for mechanism \(i\) is a map

\[ \Phi _i:(-\varepsilon ,\varepsilon )\times U\longrightarrow \mathcal M, \qquad \Phi _i(0,p)=p, \]

on an open set \(U\subseteq \mathcal M\). Its infinitesimal intervention field is

\[ v_i(p) = \left.\frac{\partial }{\partial \alpha }\right|_{\alpha =0} \Phi _i(\alpha ,p). \]

The strength \(\alpha \) is not the assigned value in a hard intervention \(\operatorname {do}(X_i=x)\). A hard-intervention family gives a smooth protocol only after we specify how it approaches the observational model. Different implementations aimed at the same substantive variable can therefore generate different fields.

Definition 5.4 Visible intervention distribution

For fields \(v_1,\ldots ,v_m\), define

\[ \mathcal D_{\mathrm{vis}}(p) = \operatorname {span}\{ v_1(p),\ldots ,v_m(p)\} \subseteq T_p\mathcal M. \]

We work on a stratum where this distribution has constant rank \(r\), with a local frame \(w_1,\ldots ,w_r\).

Diagram illustrating 5.3 Smooth intervention protocols.

None

The normal bracket component records failure of the visible distribution to close.

When intervention protocols lift to coherent endotransformations of a realized model, the 2-categorical picture gives a useful local test. The two orders of intervention are vertical composites of the same two cells. Their first non-vanishing discrepancy is represented infinitesimally by the Lie bracket. The normal component is the part for which the current visible intervention language has no wire.

Sequential interventions as vertical composition. The picture is licensed only when the finite protocols have been typed as coherent model transformations. The bracket is the infinitesimal commutator of their generated fields, not a declaration that every intervention is a… — Figure 5.1 Sequential interventions as vertical composition. The picture is licensed only when the finite protocols have been typed as coherent model transformations. The bracket is the infinitesimal commutator of their generated fields, not a declaration that every intervention is a natural transformation.