MMP.19:4.2 - Construct one common basis for the alternatives

In a structural model, write the relevant variables as functions of their parents and underlying inputs U. Recover the joint law p(u), given input values, or a set of admissible input laws and mechanisms. U can include shared disturbances and persistent characteristics; its meaning depends on the subject account.

For each action a, let Y_a(u) denote the output of the modified model at the same underlying input u. A response-type representation can instead give a joint vector such as (Y_0,Y_1) directly. Its joint distribution states which responses belong to the same modeled unit. Having the two marginal distributions alone does not supply this dependence.

Keep common conditions common, and identify which changes the action itself causes. Two runs with independently drawn U describe different modeled cases. Reusing a random seed is a numerical way to implement a specified coupling; the seed does not justify that coupling in the subject. Repeating work later with changed conditions likewise needs an account of which inputs persist and which are newly drawn.

Use a model with well-defined responses for the requested inputs. In the finite acyclic case, evaluation in dependency order suffices. Continuous, cyclic or multiple-solution models require their own existence and solution-selection conditions; a solver’s returned trace does not settle an unspecified response relation.