MMP.11:5 - Archetypal Grounding

MMP.11:5.1 - Construct a response from its known shape

A normalized input u lies in [0,1]. The subject account supports a nondecreasing response r with r(0)=0 and r(1)=1. Its intermediate shape is unknown. Begin with these properties, rather than choosing a straight line as the only candidate.

Choose an integrable h with h(u)>=0 almost everywhere and H=integral_0^1 h(v) dv>0. Define

r(u)=integral_0^u h(v) dv / H.

The endpoints follow by substitution. For u2>=u1, the difference is the nonnegative integral of h over [u1,u2], divided by H. Thus every member is nondecreasing. These curves are absolutely continuous. Every absolutely continuous nondecreasing response with these endpoints has such a representation using its almost-everywhere derivative, but a jump response is outside this family. The needed regularity must come from the question or remain a declared restriction.

For a small calculable family, use linear segments through (0,0), (1/4,q), (1/2,1/2) and (1,1). Their slopes are 4*q, 2-4*q and 1. They are nonnegative exactly when 0<=q<=1/2. This is a construction of admissible candidates, not a conclusion from measurements alone.

Suppose observations establish only the three values at 0, 1/2 and 1. Every q in that interval agrees with them. The consequence r(1/4)<=0.6 follows for this whole family; it also follows for every nondecreasing response with the given midpoint. There is no need to identify q for that question.

Now the receiving use asks whether r(1/4)>0.3. Candidates q=0.2 and q=0.4 satisfy the same observations and give opposite answers. Another repetition at the three old input values does not distinguish these ideal candidates. An observation near the disputed input may help; its precision and cost belong to that new question. Alternatively, a supported additional shape relation could narrow the family.

MMP.11:5.2 - Retain an exchange balance without inventing its mechanism

Two nonnegative amounts x and y exchange a conserved total N. The forward and reverse rates are unknown. Use locally Lipschitz nonnegative rate functions a(x,y) and b(x,y), defined on a neighborhood of the nonnegative states being used, and construct

q=x*a(x,y)-y*b(x,y),

x_dot=-q, y_dot=q.

Adding the two equations gives zero change in x+y for every admitted a and b. At x=0, x_dot=y*b(0,y)>=0; at y=0, y_dot=x*a(x,0)>=0. With these regularity conditions the continuous-time solution preserves nonnegativity. These are properties of the coupled construction. The applicability of conserved exchange to the subject remains a premise.

Even complete knowledge of q need not identify the gross transfers. For any nonnegative locally Lipschitz h, define

a_new=a+y*h, b_new=b+x*h.

The two added contributions to q are x*y*h and -y*x*h, which cancel. The whole observed evolution is unchanged. This is an algebraic family of alternatives, not merely several successful numerical fits.

For a dimensionless instance, take a=b=1. The alternative h=1 gives a_new=1+y and b_new=1+x, yet both models have q=x-y. At x=2, y=1 they both predict x_dot=-1.

Change the question: a proposed intervention suppresses only the reverse transfer while leaving the forward rate law applicable. Under that causal premise, C.28.MR replaces the reverse contribution by zero. The first model gives x_dot=-2; the second gives x_dot=-4 at the same state. Ordinary observations of x and y under the unchanged mechanisms cannot choose between these accounts. A prediction under the old operation can still be useful; the proposed intervention needs a contribution that distinguishes the mechanisms or a sufficient bound covering them.

For dimensional quantities, a and b have inverse-time units, while h has inverse-amount-inverse-time units. Restoring units prevents treating the added terms as arbitrary dimensionless corrections.

MMP.11:5.3 - Construct probabilities while keeping boundary outcomes

A report has three possible outcomes. Let w_i>=0 and let their sum W be positive. Set p_i=w_i/W. Every candidate has nonnegative probabilities summing to one. Conversely, every probability vector on these outcomes is represented by choosing w=p. Thus this construction includes zero-probability outcomes.

The weights (0,1,3) and (0,2,6) both give probabilities (0,1/4,3/4). The parameter vector is redundant even if the probability vector becomes fully determined. There is no need to distinguish those weights when the receiving question uses only the law.

A strictly positive parameterization, such as exponentiating every finite unconstrained parameter before normalization, excludes zero probabilities. It can approximate a zero closely but cannot express it with finite parameters. If the subject account rules out the first outcome, retain that zero in the construction and normalize weights for the remaining outcomes. Whether a very small nonzero value would suffice depends on the receiving question.

The constructed p is a family member, not yet an estimate from data. MMP.7 composes it with selection, rounding or other recording behavior. The appropriate statistical method then determines what the observations support. Changing the recording procedure can change that inference without changing the underlying outcome family.