MMP.11:4.2 - Construct the permitted variation

Translate each retained property into a mathematical condition, then choose a construction that satisfies it. MMP.10 supplies the general work of representing candidate objects and their conditions. Here the object being constructed is a family of relations to insert into the model. Direct conditions may already give a workable representation of that family. Constructing through free elements is useful when their variation should preserve the conditions; compare its obtaining and revision operations with those of the direct representation.

A useful construction separates a fixed part from free variation. If a linear operator L must satisfy L(g)=b, find one particular solution g0 and choose a correction h with L(h)=0. Then g=g0+h retains the condition. This describes every solution only if the admitted corrections cover the whole null space in the chosen function domain. Restricting h to a few basis functions supplies a smaller family. Establish the linearity and domain before using this construction.

For a sign, bound or shape condition, construct through a map whose output has that property. Nonnegative weights can be normalized to probabilities. Integrating a nonnegative function can produce a nondecreasing response. Work out the domain and boundary of the resulting family: strict positivity excludes zeros, and an integral of an ordinary integrable function produces an absolutely continuous curve. Section :5.1 develops one such construction and its restriction.

When a property depends on coupling, construct the coupled contribution. Using the same transfer with opposite signs in two balance equations preserves their total. Two separately fitted right-hand sides have no such identity unless their joint conditions supply it. Use A.3.3.TR to assemble a change rule from the interacting relations.

A penalty during fitting offers a different construction: it discourages violations while allowing them. Use it when that allowance fits the question. If the account requires an identity, either build it into the representation or use an obtaining method that enforces it. The size of a training penalty does not by itself establish the identity.