MMP.18:4 - Solution

Choose the combined answer. Recover the meaning and conditions of each consequential exchange. Construct interface relations that preserve what the receiving use needs, then impose them with the component models. Obtain and interpret a joint result, distinguishing a formulation defect from an approximation in its computation. Reopen only the exchange or component whose changed assumption affects that result.

MMP.18:4.1 - Identify what the models share and what they exchange

Start from the required result and trace which component supplies each quantity it uses. Recover the quantity’s referent, unit and reference point, as well as any time interval, spatial region or probability conditioning that changes its meaning.

Distinguish a shared quantity from a transferred result. If both models describe the same unknown offset, retain one common offset or a justified relation between their coordinates. If one supplies an estimated offset to the other, specify whether the receiving calculation uses its uncertainty or only an approximation by a point value.

Locate overlapping subject effects. If both component laws already include the same interaction, composing them may count it twice. Decide from the subject model which contribution is being retained and which is a second description of it. A.3.3.TR and C.29.BB supply the resulting joint change and balance relations.

Also recover the operating conditions. A component calibrated for one boundary condition may cease to apply when connected to a responsive neighbor. Replacing a supplied input by another model’s output can therefore require a changed component law, even when the variable types match.

MMP.18:4.2 - Construct the interface relation

Express the exchange as a relation between the component descriptions. A deterministic map is appropriate when the supplied result determines the needed input. Use a broader relation or conditional law when several inputs remain possible.

For a conversion, translate the quantity and its reference point. For an aggregate, state the operation over the specified region or interval. For example, an amount over [t0,t1] is the integral of a rate over that interval; a rate at t0 alone determines that amount only under an additional evolution assumption.

Let x range over values expressed by the supplying model and y over those used by the receiving model. A relation H(x,y) states the compatible pairs. A.3.3.TR supplies composition with other relations, and MMP.10 expresses the joint possibilities. MATH.18 helps determine which operations and consequences the interpretation preserves. Equality of numerical values is one possible relation, not the default.

Derive the property the map must preserve. For amounts at locations, a linear map b=W*a preserves their total for every a when every column of W sums to one. For values of an intensive field, a map preserving a constant field instead has every row sum to one. If amounts must remain nonnegative, nonnegative weights are an additional sufficient condition. The needed property chooses the map; a convenient interpolation routine does not choose the property.

The two conditions solve different problems. A common field value should remain that value on another discretization. The total amount distributed among cells should remain the same total. When both requirements matter, construct a map with the appropriate geometry or integration weights rather than substituting one condition for the other.

MMP.18:4.3 - Restore what the receiving scale needs

Apply the receiving operation to the proposed supplied description. If the result still depends on discarded detail, identify that dependency before choosing a closure.

For example, a component supplies only the mean of x over a region, while a neighboring response requires the mean of x^2. The missing contribution is the variance:

mean(x^2) = mean(x)^2 + variance(x).

Two fields with the same mean can therefore yield different received responses. Supplying the squared mean silently sets the variance to zero. Retain the needed statistic, obtain it from the finer model, supply a justified closure or return bounds sufficient for the question. MMP.9 derives reduced evolution and closures; MMP.17 constructs a surrogate when that is the chosen supplier.

A time-scale change can create memory. An interval average need not determine a response at an intermediate instant. A rapidly changing component can also leave an accumulated effect after its local state has relaxed. Recover the information needed by the receiving answer instead of assuming that a small or fast component has no relevant contribution.

Use a coupled approximation only within the conditions supporting it. If feedback drives a surrogate or closure into a different regime, revise that contribution or return to a suitable supplier. Good behavior on isolated component inputs does not establish behavior on inputs generated by the coupled system.

MMP.18:4.4 - Impose the joint conditions before choosing execution order

Combine the component relations and interface conditions for the same modeled situation. Shared boundary values and initial conditions must satisfy that joint formulation. Redundant equations can express a useful invariant; contradictory equations expose incompatible assumptions or a failed identification of the shared quantities.

Feedback can require a joint solve. Suppose the selected same-instant relations are x=1+y and y=x/2. Substitution gives x=2 and y=1. Starting from y=0 and calling the first model once, then the second, returns x=1 and y=0.5; those values fail x=1+y. That call order is only one unfinished computation of the joint relation.

Choose between a common solve and separate interacting computations from the available capabilities and error needed by the answer. For time-dependent components, specify which inputs are held, interpolated or extrapolated between communication times, how events are synchronized, and whether a proposed step can be repeated. These are computation assumptions unless the actual subject has that delay or sampling behavior.

Compare the obtained exchange against the joint conditions and the required result. A residual can reveal a mismatch. Turning a small residual into an error bound needs the relevant conditioning or stability argument; CMP.8 supplies that approximate-computation work. CMP.14 supplies interaction between computations. More iterations cannot repair an incompatible physical or statistical premise.

MMP.18:4.5 - Combine uncertainty without duplicating information

When components are probabilistic, construct a joint law for the common quantities and the component-specific quantities. Marginal distributions alone generally leave dependence unspecified. Obtain the needed dependence from the modeled mechanism, a conditional law or an explicit additional assumption.

For components sharing z, one possible factorization is:

p(z,u1,u2) = p(z) * p(u1 | z) * p(u2 | z).

It assumes conditional independence of u1 and u2 given z. If that assumption is unavailable, construct their joint conditional law or retain the unresolved dependence. A common random quantity is sampled or integrated once as that same quantity; two independent draws would describe a different model.

For two analyses using the same positive prior pi(z) and conditionally independent data D1,D2, their posteriors satisfy q1(z) proportional to pi(z)*L1(z) and q2(z) proportional to pi(z)*L2(z). The combined posterior is proportional to q1(z)*q2(z)/pi(z) on the common prior support. Multiplication without the division counts the prior twice.

If the analyses use overlapping records, first recover which observations and likelihood factors are shared. Dividing out a prior does not remove a duplicated observation. If component priors disagree, selecting a common prior or a pooling rule changes the model and needs a stated basis. MMP.7 supplies the record law and MMP.13 the resulting inference; a computational sampler obtains values from that law.

Point estimates can still be sufficient for a particular receiving use. To replace a distribution by a point, establish that the omitted uncertainty does not alter the required result at its chosen tolerance. Keep a consequential dependence when a nonlinear operation or tail probability needs it.

MMP.18:4.6 - Return the coupled result and localize a failure

Return the combined formulation with the exchanged quantities and assumptions needed to reproduce the required result. The subject recipient must be able to interpret that result, including any approximation or unresolved interface contribution that changes its use.

Check the consequence at the scope actually claimed. An interval balance establishes the amount transferred over that interval, not the time of a threshold crossing within it. A joint posterior accounts for the supplied data under its dependence assumptions; it does not establish that those assumptions describe the subject.

If a result fails, locate whether the problem lies in the quantity identification, map, closure, component law, dependence or computation. Repair that contribution and its affected consumers. For a physical question, B.5.MPC.R coordinates a change crossing the physical, mathematical and computational accounts.

Stop with the sufficient coupled answer or the missing contribution that prevents it. C.11.DUA governs whether obtaining that contribution is preferable to a restricted answer or a different method. More detailed coupling is useful when its difference matters to the work.