C.16.IR:4.2 - Construct the jointly compatible cases

Substitute the obtained indication into the relation. Combine that condition with the admitted domains and available information about influences. For repeated or multiple readings, state which subject values and parameters are shared and which may change between them.

With bounded error, include the unknown error in its stated range. If a relation gives an ideal indication f(x,z), x is the sought value, z contains influential unknowns and r is the reported reading, a common model is:

r = f(x,z) + e, with e in the supplied error range.

Use that equation only when additive error describes the actual indication convention. An error before clipping, an error after clipping and a missed event can require different relations.

A compact mathematical expression for the compatible sought values is:

S(r) = {x : there exist admitted z and e for which r = f(x,z) + e}.

An implicit relation can be used in the same way without converting it to a single-valued function. Retain dependencies among unknowns: independently combining their separate ranges can introduce cases excluded by their joint relation.

A probability model supplies a different additional contribution. Values inside a bounded-error range have no assigned likelihood merely from membership in that range. A confidence interval or posterior distribution requires its corresponding statistical method and assumptions.