MMP.12:4.1 - Construct the forward relation from the modeled situation

Name the unknown x, the target q=T(x), and the records y. The unknown may be a parameter vector, a function or a collection of relations. State its domain and the units and scales used to compare changes. A target can be a total, a value at one time or a threshold; it need not be x itself.

Follow a candidate x through the modeled process and recording operation. Write the resulting ideal record as F(x,z), where z contains influential unknowns that are not the target. Keep known inputs fixed and retain shared unknowns across records. MMP.11 constructs a missing model family; C.16.MR supplies a missing measurement relation. If probabilities matter, MMP.7 constructs the recording law.

For an additive bounded-error account, one possible formulation is

y_delta = F(x,z) + e, with ||e||_Y <= delta.

Here delta bounds error in the chosen record norm. Use this form only when the recording procedure supports additive error. For an implicit forward relation, retain its equations and jointly unknown outputs rather than forcing it into a single-valued map. A likelihood from MMP.7 can instead supply the data discrepancy appropriate to a probabilistic account.

Include consequential uncertainty in calibration and in the forward approximation. A fixed but unknown offset remains an unknown; setting it to zero can make recovery appear better determined. A noise bound and a standard deviation have different meanings and support different conclusions.