Contribution
It proposes that individuals be found rather than assumed: an individual is an aggregate that “propagates information from its past into its future”. For a system S and its environment E it decomposes the one-step predictive information about the system and reads the terms as kinds of individuality. The 2020 version adds a partial-information decomposition of those terms, a worked two-unit example and the admission that the measures cannot locate boundaries without a size penalty.
Key insight
The predictability of a system’s next state can be credited to the system first and the environment second, or the other way round. Credit the system first and the system’s own term is A*; credit the environment first and it is A. The two orders differ by the shared minus the synergistic information, so a system can score high on one and low on the other, and the difference separates a self-maintaining organism from a colony that persists only through interaction with its surroundings.
Assumptions
- A joint Markov process (S_n, E_n) with kernels φ(e, s; s′) and ψ(s, e; e′), finite state sets (2020, §“The origin of information”).
- A partition given in advance. The measures score a chosen system–environment split; finding the split is the open part.
- Consistent measurements over a time scale at which regularities show; the authors say the right scale “can [not] be known in advance”.
- A partial information decomposition whose concrete measure is left open (2020, footnote 1): “we will use the decomposition as a conceptual framework”.
Key results
- Two decompositions. I(S_n, E_n; S_{n+1}) = A* + nC = I(S_{n+1}; E_n) + A, hence A* + nC = A + I(S_{n+1}; E_n) and A* − A = I(S_{n+1}; E_n) − nC, called non-trivial informational closure (2014) and environmental coding (2020).
- PID form (2020, eqs. 2–5). A* = SI + UI_S, A = CI + UI_S, nC = CI + UI_E, NTIC = SI − CI.
- Closure implies sufficiency. Under the Markov structure, nC = I(S_{n+1}; S_{n−1} | S_n) + I(S_{n+1}; E_n | S_{n−1}, S_n), so nC = 0 forces I(S_{n+1}; S_{n−1} | S_n) = 0; closure is the stronger notion (appendix; Pfante et al. 2014 for the general case).
- Enlarging the system. Moving ΔS from E into S gives A′* = A* + I(ΔS_{n+1}; S_n | S_{n+1}) + I(S_{n+1}ΔS_{n+1}; ΔS_n | S_n) and A′ = A + I(S_{n+1}ΔS_{n+1}; ΔS_n | E′n) + I(ΔS{n+1}; S_n | E′n ΔS_n S{n+1}), so “both individuality measures can only grow or stay constant with increasing system size … Thus, they 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” (2020). nC′ can move either way.
- Example (2020, Figs. 2–3). Two ±1 units with logistic updates in self-coupling α, cross-coupling β and interaction γ. The text says A and A* coincide at γ = 0 and high α, β, diverge as γ grows (A* falls, A rises), and that a correlated environment moves information from nC into A*.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | A* + nC = A + I(S_{n+1}; E_n) | strong | chain rule, checked |
| C2 | Informational closure implies sufficiency of the present state | strong | appendix, checked |
| C3 | A* and A are non-decreasing in the system’s extent | strong | expansions, checked (THEORY-1163) |
| C4 | Boundaries need a size cost the measures do not contain | strong | follows from C3; stated in 2020 |
| C5 | A*, A and nC are three distinct forms of individuality | weak | interpretation; one two-unit example |
| C6 | Maximizing A* identifies “complex organisms in their environments” | weak | asserted; at odds with C3 unless the candidates have fixed size |
Method
None implemented. The 2014 version proposes adding variables to the system while A* or A rises and nC falls, stopping at the largest set “no better at predicting its future than the previous largest set”; the 2020 version keeps the incremental expansion but says a regularizer is needed and that finding the optimal partition is not its objective.
Concepts
- Organismal individuality A* = I(S_{n+1}; S_n); in Krakauer and Zanotto 2006, “autonomy”.
- Colonial individuality A = I(S_{n+1}; S_n | E_n); Bertschinger et al.’s autonomy.
- Environmental determination nC = I(S_{n+1}; E_n | S_n), the non-closure.
- NTIC / environmental coding = A* − A.
- Informational closure: nC = 0; the system predicts every influence of the environment on it.
Connections
The two versions are different papers in places. The 2014 arXiv v1 has the decompositions, the closure and expansion algebra and a stopping rule; it calls the two readings “genomic” and “environmental determination” and lists authors Krakauer, Bertschinger, Olbrich, Ay, Flack. The 2020 article reorders authors (Flack before Ay), renames the readings, adds the PID terms, the two-unit example, the size-cost sentence, and links to the free-energy principle and to Biehl et al.’s agent detection. Both cite Flack’s slow-variable work (LIT-1122) for coarse-graining as the route to new levels. Bourrat’s critique (LIT-005, NOTE-027) is of the 2020 measures. Kolchinsky and Wolpert (LIT-1149) cite the 2014 preprint.
Bearing on the record
- QUESTION-115. The LIT asked a reading to check A11’s claim that A* and A do not decrease as the candidate is enlarged, and the size cost the paper says is needed. Both are confirmed, in the 2020 version only: the 2014 preprint makes the monotonicity claim but offers a stopping rule instead of a size cost. The check is filed as THEORY-1163. The question’s summary, which reports A11’s reading, can be marked as read and confirmed. Proposed dated note below.
- CASE-101. The case’s arm A (common input) is where A* fails, as Bourrat argued and THEORY-424 holds. The reading adds that A, which conditions on the environment, scores the common-input arm near zero only if the common input is in E; the case should state which variables it puts in E for each arm, since all three measures are relative to that split.
- LIT-005, NOTE-027. Bourrat’s A* and A are the 2020 article’s. His objection to nC as a measure of individuality is the article’s own gloss: nC “quantifies the degree of environmental determinism”, and the article does not offer it as individuality except by its heading.
- THEORY-424. Consistent: the measures are observational mutual informations, and the article nowhere intervenes.
No instruction for machine-learning practice; nothing for the anthology.
Limitations
- No algorithm, no data. Both versions leave identification to future work.
- PID unspecified, so SI, UI and CI have no values; only the classical sums are computable.
- The example’s text over-reads the figures. A record computation of the 2020 two-unit system with a random environment and γ = 0 gives A* = A = 0.998 bits at α = 5, β = 1, but A* = 0.189 and A = 0.311 at α = β = 5 (and at α = β = 3). A and A* coincide at γ = 0 when the self-coupling dominates, not “at high values of α_S and β_S” in general; with an independent environment, conditioning on it can only raise the system’s own term.
- Slips. Both versions write the joint entropy as H(S; R) and say H(S|R) where they mean H(R|S); “Sn+” stands for S_{n+1} in the appendix; the 2020 caption of Fig. 3 reads “αE = 2βE = γE = 0” for what the text calls an environment with memory; Krakauer and Flack 2010 is listed twice.
Open questions
- With a size cost, what boundary do A*, A and nC pick in CASE-101’s three arms, and do they disagree as A12 expected?
- Which PID makes the 2020 decomposition non-negative and well defined for these processes, and do the conclusions depend on it?