MMP.12:5.1 - Recover a split whose contrast is weakly observed

Suppose two nonnegative loads x1 and x2 have exactly known total s=3. A second channel measures a small contrast:

d=x1-x2; z=0.01*d+e; |e|<=0.02.

All values are expressed in fixed normalized units. The recorded z is 0.03. The inverse relations are x1=(3+d)/2 and x2=(3-d)/2, so nonnegativity gives -3<=d<=3. The error account gives 1<=d<=5; jointly, the compatible contrasts are 1<=d<=3.

If the question concerns the total or whether d is positive, this already answers it. A simulation that needs one nominal split must introduce a selection. Suppose its stated reconstruction requirement is to limit the contribution of channel error to at most 1 contrast unit, while accepting at most one-half shrinkage toward balanced loads. This is a modeling preference for the nominal input, not additional evidence that the loads are equal.

Use the objective

J_lambda(d)=(0.01*d-0.03)^2 + lambda*d^2; -3<=d<=3.

Its unconstrained minimizer, whenever it lies in that interval, is

d_lambda=0.01*z/(0.0001+lambda).

The data-error contribution is bounded by 0.01*0.02/(0.0001+lambda). Making it at most 1 requires lambda>=0.0001. The exact-data shrinkage fraction is lambda/(0.0001+lambda); making it at most one-half requires lambda<=0.0001. These two declared requirements select lambda=0.0001.

The nominal result is d_lambda=1.5, hence (x1,x2)=(2.25,0.75). Its predicted contrast record is 0.015, leaving residual 0.015 within the supplied error bound. For this fixed strength, the interior reconstruction gain is 50, compared with 100 for direct inversion. Projection onto the admitted interval cannot increase that gain.

The bias matters. If the underlying contrast were d=1 and the error e=0.02, direct inversion would return 3 and this regularized rule would return 1.5. If the underlying contrast were d=3 with e=0, the same observed record would make direct inversion correct and the regularized rule would understate the contrast by 1.5. These are two constructed compatible cases, not an empirical accuracy comparison.

For a worst-case guarantee from these records, retain [1,3]. Its midpoint 2 has maximum absolute error 1 over that interval, while the selected nominal value 1.5 has maximum error 1.5. The midpoint is the better choice for that different criterion. Regularization is justified here by the declared response and shrinkage requirements, not by a claim that it improves every error criterion.

Changed condition. A new acquisition reports the same z=0.03 with supported error bound 0.002. The compatible interval is now [2.8,3]. Direct inversion has data-error contribution at most 0.2, already below the allowed 1. Choose the weakest penalty meeting that requirement: lambda=0 now suffices and introduces no shrinkage. It returns d=3 with the interval [2.8,3]; a receiver minimizing worst-case absolute error could instead use 2.9.

Keeping the old strength would return d=1.5 and residual 0.015, incompatible with the new error bound. The changed observation condition, not more accurate minimization of the old objective, changes the useful formulation.