Contribution
Before this paper, weak emergence (Bedau) had only qualitative degrees, and autonomy had information-theoretic measures (Bertschinger et al.) that were hard to estimate for many variables. After it, both have regression-based, graded measures with F-test significance: G-autonomy and G-emergence. It also offers an “ontologically innocent” reading of downward causation, as Granger causation from a macro variable to micro variables.
Key insight
Granger’s question, whether a series’ past helps predict another, can be pointed at a variable’s own past (autonomy), or at the relation between a macro variable and its parts. A weakly emergent macro variable is then one that its parts do predict, but not completely, so that its own history still helps.
Assumptions
- Covariance stationarity of every series (§2.1). Both simulations first-difference the series to get it, and Dickey–Fuller tests confirm it for the predator–prey data (§3.2.2).
- A regression model class: linear MVAR, or polynomial (Taylor) regressors of order q for the nonlinear measures (Eq. 5). The model order is chosen by AIC.
- The variable set is the analyst’s. G-emergence can be positive because relevant micro factors are left out, or because the macro variable is more transparent to the prediction algorithm than the micro variables are. With the system itself as the predictor it would be zero, “since … micro causes are jointly sufficient to give rise to macro properties” (§5.2).
- No temporal (diachronic) emergence. A process with new features over time is nonstationary and cannot be measured (§5.2.1).
- Every element must vary over time. Autonomy from a static feature of the environment cannot be measured (§5.1).
Key results
- Eq. 3, G-autonomy — ga_{X1|X2} = log(var(ε₁ with X1’s own lags omitted)/var(ε₁ full)). Significance by an F-test that X1’s own coefficients are zero.
- Eq. 4, G-emergence — ge_{M|m} = ga_{M|m}·(1/N)Σᵢ gc_{mᵢ→M}. It is zero if M is independent of m (no G-causation) or fully predicted by m (no G-autonomy).
- Eq. 7, Table 1, a constructed five-variable system — the G-autonomies sort as built: white noise X1 and X1-driven X2 score zero, while X3 and X5, with their own autoregressive terms, score high. X4’s own-past predictability is carried by X3, so its score is low. The three built-in G-causalities are the significant ones. So the set of G-causalities does not fix the G-autonomies.
- §3, predator–prey — mean predator G-autonomy, linear and nonlinear: random movement is highest, then the evolved controller (condition 4), then environment plus internal state (condition 3), then pursuit of one prey (condition 2). All pairwise differences are significant after Bonferroni correction (Fig. 3). Ten trials of 10⁴ steps, order-4 models, robust to order 8.
- §4, boids — 10 boids, 25 runs of 5000 steps per condition. Linear and nonlinear G-emergence of the centre of mass are significantly higher in the hand-chosen flocking condition H than in rigid-formation L or random R (Fig. 5). Over an 11 × 11 × 11 parameter sweep the measure mostly changes smoothly, with some sharp transitions (Fig. 6). Granger causation from the centre of mass to each boid is also highest in H (Fig. 7).
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | G-autonomy captures autonomy as self-determination: zero for white noise and for purely driven variables | moderate | Eq. 7 construction and Table 1; by design |
| C2 | Evolutionary adaptation increases G-autonomy in the predator–prey model | moderate | Fig. 3; one model, one evolved controller, conditions ordered as built |
| C3 | High G-emergence tracks visually compelling flocking | moderate | Fig. 5; condition H chosen by hand as the compelling one |
| C4 | G-emergence is robust, changing smoothly over most of parameter space | weak | Fig. 6, by inspection |
| C5 | Downward causation can be measured, without ontological cost, as Granger causation from macro to micro | interpretive | §4.2; a definition, not a finding |
| C6 | G-emergence can be positive with every micro factor measured, because of the predictor’s limits | assertion | §5.2; stated as a requirement for the measure to be useful |
| C7 | Weak emergence, not strong, is the right frame for consciousness; volition might correlate with downward G-causality in the brain | speculative | §5.3; a hypothesis, untested here |
Method
- Preprocess each series to one variable (distance from the arena’s centre), drop transients, de-mean and first-difference.
- Fit a full MVAR (or polynomial-regressor) model of order p by OLS, AIC for p.
- G-autonomy: refit without the variable’s own lags; take the log variance ratio. G-causality: refit without the other variable’s lags.
- G-emergence: Eq. 4, or Eq. 6 in the nonlinear case. It is significant if the G-autonomy and G-causality terms are.
Concepts
- G-autonomy — how much a variable’s own past improves its prediction given a set of external variables (§2.2).
- G-emergence — a macro variable both G-caused by and G-autonomous from its micro variables, measured by Eq. 4 (§2.3). A “continuous version” of Bedau’s weak emergence, based on prediction, not on simulation or explanation (fn. 1).
- Downward (Granger) causation — G-causality from a macro variable to its micro variables (§4.2).
Connections
- Builds on Granger (1969, LIT-1173), Bedau’s weak emergence, and Bertschinger et al. (2008) on autonomy. It compares itself in prose with Shalizi and Moore’s predictive efficiency, Polani, Atmanspacher’s contextual emergence and Bar-Yam’s multiscale variety (§5.2.3). Transfer entropy is “a more general version of Granger causality” (§5.1.1). By Barnett, Barrett and Seth (NOTE-934) the two are equivalent for Gaussian data.
- The causal-emergence readings. NOTE-726 relays Hoel et al.’s complaint that earlier measures, Seth’s among them, were not based on causal models. That is right, and Seth says so himself (§2.1, §5.2). NOTE-791 reads Barnett and Seth’s dynamical independence, a later criterion from the same group. It asks only that the micro history add nothing to the macro variable’s self-prediction, T(X → Y | E) = 0, where G-emergence asks for dependence and autonomy at once. NOTE-010 lists Seth 2010 as a named alternative to ΦID.
- THEORY-497 says a macro feature can supervene on the present state and still carry information about the future, “because supervenience is instantaneous and prediction runs across time”. That is the gap G-emergence measures. It is my connection; neither text makes it.
Bearing on the record
- Filed as THEORY-703, Proposed: the measure behaves as designed in two hand-built models, and nothing tests it against a criterion independent of its construction.
- The centre of mass is a function of the boids’ positions at the same time. An intervention on it that leaves the boids at that time unchanged does not exist. So the downward Granger causation of Fig. 7 cannot be an interventional effect in Eichler and Didelez’s sense (NOTE-941), with or without every boid measured. Seth claims no more than prediction, and CLAIM-334 files the point so that the stronger reading has an address. The argument is mine.
- For the Anthology of the SOTA: no instruction for ML practice.
Limitations
- Both validations are hand-built. The conditions are ordered by construction, and condition H is “selected by hand” as compelling flocking. Nothing scores emergence independently of the measure.
- Each position is collapsed to distance from the arena’s centre, a nonlinear map, before any regression. For the linear measures, part of what Granger causation from the centre of mass to a boid detects may be this transformation, which a linear regression on the boids’ transformed pasts cannot undo. The text does not say which variables the Fig. 7 regressions condition on. This is my reading, not tested.
- Eq. 4 at its limits: if m’s past predicted M exactly, both residual variances in ga would be zero, and the ratio would be undefined, not zero. “Fully predicted” can only mean that M’s own past adds nothing. My reading.
- The Appendix’s genotype count does not add up: a 29-element genotype is given as five biases and “the remaining 23” weights. The network it describes, 6–3–2 fully connected, has 24 weights. A typo, most likely, with no bearing on the results.
- Linear and nonlinear G-autonomy and G-emergence agree qualitatively throughout, which Seth reads as evidence that linear measures suffice. It is equally what one would see if the Taylor terms added little in these models.
Open questions
- Does G-emergence track anything that an independent criterion of weak emergence picks out, outside models built to show it?
- How does G-emergence relate to dynamical independence (NOTE-791) and to ΦID’s causal decoupling (NOTE-010) on the same system?
- G-autonomy against autonomy as organizational closure (Seth’s own, §5.1).