Source
Krakauer, D., Bertschinger, N., Olbrich, E., Flack, J. C. & Ay, N. (2020), LIT-1137, §“Individuality measures in an illustrative example”, read in NOTE-1400. The 2014 preprint states the monotonicity too but offers a stopping rule instead of a size cost.
What was actually shown
Let the world be split into system S and environment E, and move a part ΔS of E into the system, leaving E′. Writing S′ = (S, ΔS):
- A (organismal).* A′* = I(S′{n+1}; S′n) = I(S′{n+1}; S_n) + I(S′{n+1}; ΔS_n | S_n) = A* + I(ΔS_{n+1}; S_n | S_{n+1}) + I(S′_{n+1}; ΔS_n | S_n). Both added terms are conditional mutual informations, hence ≥ 0.
- A (colonial). A = I(S_{n+1}; S_n | E′n, ΔS_n), and A′ = I(S′{n+1}; S_n, ΔS_n | E′n) = I(S′{n+1}; ΔS_n | E′n) + I(S′{n+1}; S_n | E′n, ΔS_n) ≥ I(S{n+1}; S_n | E′n, ΔS_n) = A, the inequality because adding ΔS{n+1} to the first argument cannot lower a conditional mutual information.
So along any chain of nested candidates both measures are non-decreasing, and the whole world, with an empty environment, attains the maximum of each. The article draws the consequence itself: the measures “are not sufficient to detect the precise boundaries between individuals. In order to obtain precise boundaries we would need to impose a cost function—or regularizer—on system size”. The third measure, nC = I(S_{n+1}; E_n | S_n), changes by −I(S_{n+1}; ΔS_n | S_n) + I(ΔS_{n+1}; E′n | ΔS_n, S{n+1}, S_n) and can rise or fall.
What could have come out otherwise: a conditional measure can fall when its condition grows, so A was not monotone by inspection. It is because the variables moved out of the condition enter the arguments.
What this does not say
- It does not say the measures are useless. Compared across candidates of equal size, or normalized (Bourrat’s C1 divides by an entropy, NOTE-027), they can still rank.
- It does not supply the size cost, or say what form it should take. The record’s priced individuation (CLAIM-154, CLAIM-157) is one candidate, untested against these measures.
- It does not say nC finds boundaries. nC can fall as the candidate grows, but it also reaches zero for the whole world, so minimizing it has the same end point.
- It says nothing about whether the measures track propagation rather than association; that is THEORY-424.