MMP.9:4.3 - Choose and construct a useful replacement

Use the expression just derived to decide which information or calculation earns its cost.

Retain a quantity that obtains the missing contribution. If c depends on a small additional observable v, derive v’s update from the source law and repeat the closure test for the pair (x,v). This operation can reveal a useful finite system or a growing hierarchy. For members obeying Z_dot=-Z^2, the mean m has m_dot=-mean(Z^2). Retaining q=mean(Z^2) gives q_dot=-2*mean(Z^3). The new equation exposes the next assumption needed; adding q alone does not finish the closure.

Compute the effect through memory. Use the history expression from :4.2. To truncate old history, bound its omitted contribution for the admitted histories. To replace the memory kernel by a simpler one, bound or estimate the resulting difference on those histories, then propagate that difference through the retained evolution. A slowly decaying kernel can make distant history consequential. An auxiliary-state representation can be cheaper than truncation when a few modes express that kernel.

Bound the requested output. If the use asks for a threshold or interval, derive bounds directly from the source law and available initial information. For a population, solve or bound the member response as a function of its initial value, then average using known ranges or moments. Monotonicity, convexity or conservation can give a bound without choosing a complete distribution. The nonlinear example in :5.2 constructs such bounds. Use C.29.1:4.5 to turn them into the corresponding decision, or to expose the still-unresolved range.

Approximate a contribution whose accumulated effect is small. Identify the parameter and the class over which smallness is claimed: initial values, inputs, horizon and required error. Derive or bound the contribution integrated over that horizon. A small coefficient can multiply a large hidden value, and a fast transient can shift the later state. If a transient only matters near the start, retain its effect in an adjusted initial value and state when the later approximation begins. :5.3 shows both constructions.

A learned closure is another possible approximation to the missing contribution. Specify what its inputs contain and how its output enters the retained update. Pairs of retained inputs and source contributions can support fitting, but different hidden states can give different contributions at the same retained input. A fitted conditional mean then answers a distribution-dependent question; it is not an all-cases replacement of those contributions. When probability is needed, MMP.7 helps construct the law of the sampled or recorded training cases. Use the resulting closure at its intended inputs and horizon before drawing the corresponding predictive conclusion.

Compare the candidates through the existing characterization and choice methods: the required output or bound, obtaining cost, initial information and conditions. There is no need to develop every alternative. C.11.DUA helps choose whether more derivation, data or computation can change the decision.