Contribution
A unification and a demonstration. The unification writes Galton’s covariance, Eells’s and Suppes’s probability raising, Cheng’s causal power, Lewis’s relative risk (also with a closest-possible-world counterfactual), Pearl’s PN, PS and PNS, a bit-flip sensitivity, effect information and EI in terms of two “causal primitives”. The demonstration computes the difference between macro and micro values of each on one model and finds positive differences for every measure somewhere in its parameter grid.
Key insight
Every measure compares P(e | c) with what happens to e without c, and that second term needs a distribution over causes. Coarse-graining a noisy micro chain onto a deterministic macro raises sufficiency and necessity of the macro transitions, so any measure built from them can rise at the macro. The authors read this as error correction by the macroscale.
Assumptions
- Causes and effects are states of a finite Markov chain; P(e | c) is the transition matrix. P(C) must be supplied: the stationary (“observational”) distribution, the uniform (“maximum entropy”) one, or a uniform distribution on a Hamming ball of radius Δ around the actual state (“local”).
- The model (§4.1, §7.1). 2^N states in two groups A and B; each state moves only to the other group, to a main effect with probability p and to the rest of the group uniformly. Determinism is varied by p, degeneracy by an algorithm that stacks main effects onto fewer targets. The macro is fixed as {A, B} with transition matrix [[0, 1], [1, 0]], “independent of the model parameters”. The main text uses 16 states, 8/8; Figure S3 adds a 13/3 version.
- In the code the macro matrix and its distributions are set directly, uniform over the two macro states, not derived from the micro (compute.py: “for macro tpm this should be obtained by summing the microtpm, but for the bipartite model it doesn’t matter”).
- Averages over all transitions weight by P(c, e) = P_obs(c)P(e | c), the stationary distribution, whatever distribution is used for P(C) inside each measure (Figure 4 and S3 captions; compute.py).
Key results
- Identities (§3). PNS = Eells’s measure, PN = Lewis’s normalized relative risk, PS = Cheng’s causal power, under exogeneity and monotonicity. Galton’s covariance is P(c)P(C\c)[suff + nec − 1]. Effect information is log₂ of Suppes’s ratio, and EI is its expectation. These are definitional rewritings, checked by hand.
- Sensitivity (§4.3, Fig. 3). Most measures rise with determinism and fall with degeneracy; the bit-flip measure is the exception.
- Emergence (§5.2, Fig. 4, S3). “All measures of causation exhibited instances of causal emergence”, most in secondary transitions and in noisy, degenerate regimes, with “super causal emergence” where a micro transition is preventive and its macro positive. “The only condition to not show causal emergence was the overall effective information when using the observational distribution. This was known [20].”
- The record’s check. In the symmetric model the uniform micro distribution maps to the uniform macro one and the stationary micro distribution to the stationary macro one. So these are common distributions, and the system-level EI cannot exceed the micro’s (THEORY-1167). Computed: micro EI 1.043, 1.783 and 3.326 bits at determinism 0.1, 0.5 and 0.9 with injective main effects, and exactly 1.000 with fully degenerate ones, against the macro’s 1 bit, under both distributions. In the 13/3 model the paper’s uniform macro distribution is not the image of the uniform micro one, (13/16, 3/16). There the macro wins on EI under the paper’s choice (1.000 against 0.716 at determinism 0.1) and loses under the image (0.696).
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | Popular measures of causal strength are functions of sufficiency and necessity, or determinism and degeneracy | strong as rewriting | §3 algebra; the bit-flip measure excepted |
| C2 | Every measure examined shows causal emergence on the bipartite model | moderate | Figs. 4 and S3, one hand-built model |
| C3 | Causal emergence is “generally unaffected by choice of intervention distribution” | weak | the same model; EI under the stationary distribution is the exception; no other system |
| C4 | Macroscale causation “is a general fact about causal relationships” | weak | extrapolation from C1 and C2 |
| C5 | EI is the most conservative measure of emergence | moderate | the model; it is the one measure bounded by data processing |
| C6 | Local intervention distributions are a good compromise | weak | advocacy, with the Hamming metric depending on state labels |
Concepts
- Causal primitives: sufficiency suff(e, c) = P(e | c); necessity nec(e, c) = 1 − P(e | C\c).
- Determinism / degeneracy coefficients: 1 − H(e | c)/log₂ n and 1 − H(e | C)/log₂ n, at system level averaged over P(c).
- Intervention distribution: P(C), the distribution over which counterfactuals are taken; observational, maximum entropy, or local.
- Causal emergence: CE = CS_macro − CS_micro for a measure CS; negative is causal reduction.
Connections
It extends Hoel, Albantakis and Tononi 2013 (LIT-1038, THEORY-222) from EI to other measures, and cites Hoel 2017 (not registered) as the source of the observational-distribution concession. It answers Aaronson’s and Dewhurst’s criticisms. It does not cite Eberhardt and Lee (LIT-1839), which appeared a month later. It cites Varley and Hoel (LIT-027) for synergistic information growing at macroscales, and Rosas et al. (LIT-1244) and Mediano et al. (LIT-025) as related approaches. The measures’ catalogue follows Fitelson and Hitchcock (not registered). CE 2.0 (LIT-1136, NOTE-1387) later builds on the same primitives.
Bearing on the record
- THEORY-787’s promote_when. The triage expected that the paper would either meet the promote_when or show itself to be more toy systems. It is the second. Its only system is a bipartite chain built so that its macro is a deterministic NOT gate “no matter the underlying noise”. That is a hand-built toy with positive CE, which the promote_when says cannot settle it, and nothing in it is an independent interventionist test of which level the effects track.
- THEORY-787’s “in trouble if” clause and QUESTION-204, first half. The paper concedes the decisive case without naming it. Under the system’s own stationary distribution, used at both levels, EI shows no emergence, “This was known”. The record’s derivation shows the concession is general: under any one distribution over micro states, macro EI ≤ micro EI (THEORY-1167). The paper’s reply, that other measures and individual transitions still show emergence, changes the measure, not the distribution. Those measures are pointwise or normalized by log₂ n, and nothing bounds them by data processing. So the reply is not evidence for EI’s level, which is what THEORY-787 asserts.
- A sentence that cannot be right as written. In the symmetric model, system-level EI cannot show emergence under the maximum-entropy distribution either, since that distribution is common to both levels there. If Figure S3’s maximum-entropy EI panel shows emergence, it comes from the transition averages, which weight by the stationary distribution while taking P(e | C) from the uniform one, or from the 13/3 column, where the macro distribution is changed. The figure was not read for numbers, so which one it is stays unsettled.
- THEORY-222. Nothing to change. The arithmetic it holds is the arithmetic this paper extends.
No instruction for machine-learning practice; nothing for the anthology.
Limitations
- One model, built so the macro is deterministic and consistent at every parameter. The asymmetric variant changes only the group sizes.
- The macro is posited, not found by a search, so there is no question of which level a measure would pick; positive CE for one given partition is all that is shown.
- The averages mix distributions: P(e | C) under the chosen P(C), weights under the stationary one. So “EI under the maximum-entropy distribution” in the figures is not the EI of §3.10.
- The Hamming metric behind the local distribution and the bit-flip measure depends on the arbitrary binary labelling, as the authors note for the bit-flip measure.
- Errors and slips. In §3.1 the covariance derivation’s middle lines carry +P(c)P(C\c)P(e | C\c) where a minus is needed; the last line is right. The system degeneracy is written Σ_e P(e | c) deg(e), where P(e | C) is meant, and the effect information log₂ n[det(e, c) − deg(c)], where deg(e) is meant. §3.10 points to “Section 3.4” for intervention distributions, which are in §4.2. §7.1’s secondary probability (1 − p)/N_B does not normalize; the code uses (1 − p)/(n_B − 1). The abstract’s “we prove” covers the rewritings of §3, not a theorem that emergence occurs.
Open questions
- Does any measure that the data processing inequality bounds show emergence under a common distribution? If none does, “widespread across measures” is a property of the unbounded ones.
- On a system not built for it, do these measures agree on which partition is best, once the partition is searched rather than posited?