NOTE-791: Barnett and Seth — Dynamical independence

Contribution

The paper defines when a coarse-grained variable is a dynamical system “in its own right”. It is when knowing the microscopic history does not improve that variable’s prediction of itself. It turns this into a measure: transfer entropy from micro to macro, or Granger causality in the linear case. The measure is invariant under reparametrization and transitive across scales. For linear state-space systems it is given in closed form, and minimizing it over all linear coarse-grainings yields the most independent macroscopic variables at each scale. Before this, related criteria (informational closure, G-emergence, causal emergence, Markov-preserving partitions) either needed a candidate supplied in advance or lacked one of these properties.

Key insight

An emergent variable is a way of looking at the system that predicts itself as well as the whole microstate predicts it. In a causal graph, that is a sub-graph with no incoming information transfer. It may send influence out, but receives none. Dynamical independence does not require the variable to predict itself well. White noise can qualify.

Assumptions

• A given system/environment boundary. “Here we assume that the system/environment boundary is given” (citing Krakauer et al. 2020 for that problem).
• Supervenience as a predictive condition (2.1): I(Y_t : Y_t⁻ | X_t⁻, E_t⁻) ≡ 0. Any deterministic coarse-graining Y_t = f(X_t) satisfies it.
• Admissible coarse-grainings (2.8). f must have a complement u with f × u an isomorphism. This holds for linear maps and discrete-state systems, and is needed for transitivity.
• Stationarity. Strong stationarity, so the time index is dropped.
• The linear analysis (§3).
  • X is wide-sense stationary, purely nondeterministic, stable and minimum-phase. It has VMA and VAR forms, and the innovations need not be Gaussian.
  • Coarse-grainings are linear, Y = LX with L full rank m × n, so the candidates are points of the Grassmannian G_m(n), not partitions of the units.
  • TE equals GC exactly for Gaussian innovations. Otherwise GC is used as a linear-prediction measure in its own right.
  • The environment is set empty “for simplicity”. The conditional spectral case is left “for a future study”.
• The scale m is chosen. The search is run separately at each m = 1, …, n − 1.

Key results

• Definition (2.3)–(2.6). Y is dynamically independent of X given E iff T(X → Y | E) = I(Y_t : X_t⁻ | Y_t⁻, E_t⁻) ≡ 0. Since (2.1) holds, this equals H(Y_t | Y_t⁻, E_t⁻) − H(Y_t | X_t⁻, E_t⁻).
  • The measure is the dynamical dependence T(X → Y | E).
  • It is invariant under isomorphisms of source, target and conditioning space (2.7).
  • T(Y → X | E) ≡ 0 trivially, read as “the non-existence of ‘downward causation’”.
• Transitivity (2.10, App. A). If Y = f(X) is independent of X and Z = g(Y) is independent of Y, then Z is independent of X. So the independent variables form a partial order and can nest across scales.
• Causal-graph form (2.11)–(2.13).
  • After a change of coordinates making Y the first m coordinates, T(X → Y | E) = 0 iff G_ij(X | E) = 0 for i ≤ m < j.
  • Here G_ij = T(X_j → X_i | X_[ij], E) is the conditional causal graph.
  • So a dynamically independent variable is “a sub-graph of the causal graph with no incoming information transfer from the rest of the system”. Several such variables cannot generally be put in that form at once.
• Closed form for state-space models (3.16)–(3.18).
  • With Σ = I and LLᵀ = I, F(X → Y) = log|LVLᵀ|, where V = CPCᵀ + I and P solves the reduced DARE (3.17).
  • The spectral form is F(X → Y) = (1/2π)∫₀^{2π} log|LS(e^{−iω})Lᵀ| dω (3.21).
  • There is a reduced DARE for VAR(p) (3.23)–(3.24).
• Exact independence is generically rare (§3.3).
  • For a state-space model, F ≡ 0 ⇔ LCA^kKMᵀ ≡ 0 for k = 0, …, r − 1 (3.25). That is rm(n − m) quadratic equations in m(n − m) unknowns.
  • Generically there are solutions at every scale if r = 1. There are none if r ≥ n. For 1 < r < n there are solutions only in stated ranges of m.
  • For a VAR(p), F ≡ 0 ⇔ LA_kMᵀ ≡ 0 for k = 1, …, p (3.26). “Generically, for p = 1 we should again expect to find dynamically-independent macroscopic variables at all scales … while for p > 1 we don’t expect to find any”, except on a measure-zero set.
  • The spectral measure is analytic in z, so if it vanishes on any band it vanishes everywhere.
• Search (§3.5).
  • Minimizing F over the Grassmannian has no tractable analytic solution (App. C). The cost “appears to be in general multi-modal”.
  • Local optima are “of interest in their own right, as an integral aspect of the emergence portrait”.
  • A (1+1) evolution strategy with restarts works up to n ≈ 20, or about 100 with a proxy-cost pre-optimization (3.28).
• Simulation (§3.6, Figs. 1–6).
  • A 9-variable VAR(7) was built from three fully connected modules with two inter-module links, and 100 runs were made per scale.
  • Exactly independent subspaces appear only at m = 2 and m = 6. They are the sub-graphs {1, 2} and {1, …, 6}, “only these sub-graphs have no incoming Granger-causal connections”, and the first is nested in the second.
  • Other runs end in distinct local optima, some “almost orthogonal” to the global one.
  • The authors caution that the match to sub-graphs holds “as a direct consequence of its construction from a causal graph; we are, in effect, working in a ‘privileged’ coordinate system”.
• Deterministic systems (§4, App. D).
  • Shannon measures fail for continuous deterministic dynamics. For flows, f is defined as independent iff f(ξ(x, t)) = η(f(x), t) for some flow η.
  • The general solution is f = Ψ(τ(x), u(x)), with u a set of m − 1 invariants of the flow and τ a “time-like” function (D.10). It is built from conserved quantities, which links it to Noether.
  • There is no measure of near-independence for flows yet.
• Not autonomy (§5.2). In X₁,t = aX₁,t₋₁ + bX₂,t₋₁ + ε₁,t, X₂,t = ε₂,t, the white noise Y = X₂ is dynamically independent. Adding cε₁,t₋₁ to X₂ leaves it white but no longer independent.

Claims

id	claim	strength	support
C1	Dynamical dependence is transformation-invariant and dynamical independence is transitive	strong	(2.7); proof in App. A
C2	Independence is equivalent, in suitable coordinates, to a causal sub-graph with no incoming transfer	strong	(2.11)–(2.13), proof sketched by reference to App. A
C3	For linear state-space models dynamical dependence has a closed form and a spectral form	strong	(3.16)–(3.21)
C4	Generically, exact independence exists at all scales for VAR(1) and at none for VAR(p > 1)	strong as a counting argument	(3.25)–(3.26); equation and unknown counts, not a full proof of existence of real solutions. (Annotated 2026-10-19: the p > 1 half is proved in THEORY-733; the p = 1 half is false over the reals, e.g. a plane rotation.)
C5	Optimization finds the planted independent variables, and the cost is multimodal	moderate	one 9-variable toy model with 100 restarts per scale
C6	The criterion is intrinsic to the dynamics, unlike observer-query abstractions	moderate	argued against Millidge (2021), §5.1; the boundary and scale remain inputs

Method

1. Fit or specify an innovations-form state-space or VAR model of the microscopic data.
2. Normalize Σ = I.
3. For each scale m, minimize log|LVLᵀ|, or the spectral integral, over orthonormal L with a (1+1)-ES and multiple random restarts. Optionally pre-optimize the proxy cost Σ_k ‖LCA
4. Stop at a threshold, or on failure to reject zero GC for VAR models.
5. Map the optima by principal-angle distances to each other and to the coordinate axes and planes.

Concepts

• Macroscopic variable. Any process that does not predict itself beyond what the micro history and environment predict (2.1). Typically a coarse-graining f(X).
• Dynamical independence / dependence. The condition T(X → Y | E) ≡ 0, and the measure T(X → Y | E).
• Emergence portrait. The set of maximally independent variables at every scale, with their dependence values and their relations (ansatz 2.14).
• Informational closure (Bertschinger et al.). T(E → Y) = 0: the same condition, taken against the environment rather than the micro.
• G-emergence (Seth 2010). G-autonomy plus T(X → Y). For coarse-grainings it reduces to the dependence, so it “pulls in the opposite direction”.

Connections

The paper positions itself against Bedau’s weak emergence, Bertschinger et al.’s informational closure, Seth’s G-emergence, Rosas et al.’s causal emergence (built on PID, which lacks a canonical form and transformation invariance), Allefeld et al.’s Markov-preserving partitions, Shalizi and Moore’s causal states, Hoel et al.’s effective information (which needs a uniform prior over states) and Millidge’s query-driven abstractions.

Bearing on the record

• QUESTION-065. LIT-979 is registered there for “a coarse-graining fixed by the system rather than by an analyst’s target”, bearing on uniqueness. The reading confirms half of this and qualifies the rest.
  • Confirmed: a system-supplied target. The criterion fixes the target as the coarse variable’s own future, so no outside query is chosen. The authors draw exactly this contrast with Millidge, whose abstractions are “driven by the agenda of the observer”. So the paper supplies one answer to the question’s first part: yes, a target can be supplied by the system.
  • It is not QUESTION-065’s target. The question’s target is informativeness about the far environment across a buffer, which is locality. Dynamical independence is closure against the system’s own remaining degrees of freedom, with no spatial structure and no buffer. The paper says nothing about locality.
  • The analyst still fixes three things:
    • the system/environment boundary (“assume[d] … given”);
    • the scale m;
    • the class of admissible maps. In the worked theory these are linear subspaces, which mix units, not partitions of units.
  • Uniqueness: no. The cost is “in general multi-modal”. The authors keep every local optimum as part of the portrait. Exact independence is generically absent for VAR order above 1. In the one example, the exact optimum at each scale is unique because it was planted as a causal sub-graph.
  • Filed as THEORY-257, which answers both parts of QUESTION-065 in part.
• CLAIM-152. The paper is a case of an individuation by sufficiency (a coarse variable sufficient for its own future) whose target is fixed by the dynamics. It does not refute CLAIM-152, because the target is still fixed: self-prediction rather than some other outcome. It does show one target that needs no choice beyond scale and boundary.
• CLAIM-157. Near-decomposability asks for weak couplings across the block’s edge in both directions. Dynamical independence asks only that none come in. A module that drives the rest of the system and receives nothing is exactly independent, as {1, 2} is in the toy model. A module that is driven by the rest never is. The two criteria pick different blocks whenever coupling is one-way.
• CLAIM-167. On this criterion a one-way drive does not make one system of driver and driven at a fixed scale. The driver alone is emergent, and the driver-plus-driven may be a larger nested one ({1, …, 6} contains {1, 2}). That is a definite reading of “the direction of coupling gives the system’s internal architecture”, which the paper states without reference to the record.
• THEORY-198 and THEORY-205. Those select coarse variables by information about a far environment. This one selects them by closure. On a lattice the two could be compared directly. The paper does not compare them.
• No instruction for machine-learning practice. The authors’ intended application is neurophysiology. Nothing for the anthology.

Limitations

• Mostly linear, stationary and environment-free. The conditional spectral case and nonlinear coarse-grainings are left open.
• The optimizer is heuristic, and scale is limited. n ≈ 20, or about 100 with pre-optimization. The number of local optima grows with n.
• No sampling distribution for the state-space estimator. Inference is practicable only for VAR models.
• One toy example, which is privileged by construction.
• The deterministic-flow theory is preliminary (“in preparation”) and has no measure of near-independence.
• The authors’ caveat: whether emergent structures are “functionally relevant” is “an empirical question”.

Open questions

• In a local system, does a variable chosen for informativeness across a buffer have low dynamical dependence, and the reverse? This is the comparison QUESTION-065 needs, and either answer would bear on CLAIM-152.
• Does any criterion select one optimum where this one gives many? Or is multiplicity a property of the systems?
• How does the portrait change when the system/environment boundary is moved? This is the input the paper takes as given.