THEORY-764: For a deterministic supervenient feature V of a Markov system, Rosas et al.'s practical emergence criterion Ψ equals the parts' whole-minus-sum information about V's next value minus V's one-step dynamical dependence I(V_t′; X_t | V_t), so dynamical independence raises Ψ but neither implies emergence nor is implied by it, and the paper's sufficient test for causal decoupling can pass a feature that is not dynamically independent

Source

Rosas, Mediano, Jensen, Seth, Barrett, Carhart-Harris and Bor (2020), LIT-1244, read in full in NOTE-987: the definitions of supervenience (Definition 1) and of the criteria Ψ, Δ, Γ (Eq. 10), and Proposition 1. Barnett and Seth (2023), LIT-979, read in NOTE-791, for dynamical independence. Neither paper states the identity below. It is the record’s derivation.

What was actually shown

Setting. X_t = (X¹_t, …, Xⁿ_t) is a Markov process, and V_t = F(X_t) is a deterministic feature, as in all three of the paper’s case studies. With t′ the next time point, the paper’s criterion at order 1 is Ψ(V) = I(V_t; V_t′) − Σ_j I(Xʲ_t; V_t′). Write

• W(V) := I(X_t; V_t′) − Σ_j I(Xʲ_t; V_t′), the whole-minus-sum information of the parts about V’s next value;
• T₁(V) := I(V_t′; X_t | V_t), the one-step dynamical dependence.

The identity, Ψ(V) = W(V) − T₁(V).

1. Since V_t = F(X_t), H(V_t | X_t) = 0, so I(V_t; V_t′ | X_t) = 0.
2. By the chain rule, I((X_t, V_t); V_t′) = I(X_t; V_t′) + I(V_t; V_t′ | X_t) = I(X_t; V_t′).
3. By the chain rule the other way, I((X_t, V_t); V_t′) = I(V_t; V_t′) + I(X_t; V_t′ | V_t).
4. So I(V_t; V_t′) = I(X_t; V_t′) − T₁(V). Subtracting Σ_j I(Xʲ_t; V_t′) from both sides gives the identity. For order k, the sum runs over the groups of k parts and the argument is unchanged.

Checked on five random three-bit Markov chains with random three-valued features: both sides agree to 10⁻¹⁰.

T₁ = 0 implies Barnett–Seth dynamical independence. Their condition is I(V_t′; X⁻_t | V⁻_t) = 0, with histories up to t (THEORY-257, Eq. 2.6, no environment).

1. X is Markov, so p(V_t′ | X⁻_t) = p(V_t′ | X_t).
2. T₁ = 0 means p(V_t′ | X_t) = p(V_t′ | V_t).
3. So p(V_t′ | X⁻_t) depends on X⁻_t only through V_t, which is a function of V⁻_t. Hence V_t′ ⫫ X⁻_t | V⁻_t.

More precisely, for Markov X the multi-lag dependence never exceeds T₁. Write V⁻_t = (V_t, V_past). Then I(V_t′; X_t | V⁻_t) = T₁ + I(V_t′; V_past | X_t, V_t) − I(V_t′; V_past | V_t). The middle term is 0 by the Markov property, so Barnett and Seth’s dependence equals T₁ − I(V_t′; V_past | V_t) ≤ T₁. The converse implication is not claimed: V’s longer history can remove dependence that V_t alone leaves.

Consequences.

• For a dynamically independent feature (T₁ = 0), Ψ = W. Among features with the same W, Ψ is largest for the independent ones.
• Ψ > 0 does not need independence. Take three bits whose parity is kept with probability γ(x¹), with γ(0) = 0.95 and γ(1) = 0.6, the next state otherwise uniform within the parity class. Example 1 of the paper is the case of constant γ.
  • Exactly: Ψ = 1 − H(0.775) = 0.231 bits, Γ = max_j I(V_t; Xʲ_t′) = 0 and T₁ = H(0.775) − ½[H(0.95) + H(0.6)] = 0.141 bits.
  • So the parity meets Proposition 1’s stated sufficient condition for causal decoupling (Ψ > 0, Γ = 0), and it is not dynamically independent.
  • With constant γ (Example 1), T₁ = 0: the paper’s own “statistical ghost” is exactly dynamically independent.
• Independence does not give emergence. Take two independent bits, each copying itself, with V = X¹. Then T₁ = 0 and Ψ = 0. By the paper’s Lemma 1, a function of a single part is never causally emergent.
• Both can hold with no interaction. Take n ≥ 2 independent stationary Gaussian AR(1) channels with a common coefficient a, 0 < |a| < 1, and V their sum.
  • V_t′ = aV_t + noise, so T₁ = 0.
  • Ψ = −½ log(1 − a²) + (n/2) log(1 − a²/n). That is positive because log(1 − x) is strictly concave and vanishes at 0, so n log(1 − a²/n) > log(1 − a²).
  • For a = 0.9 and n = 2, Ψ = 0.449 bits; a simulation of 2 × 10⁵ steps gives 0.454.
  • The discrete analogue: two self-copying bits with V = X¹ ⊕ X² give Ψ = 1 bit and T₁ = 0.

What could have come out otherwise. Ψ could have been unrelated to closure, with an independent feature penalized or a dependent one rewarded. The identity shows the dependence term enters with a fixed negative sign and unit weight.

What this does not say

• Nothing about the PID quantities. The identity concerns Ψ, a redundancy-free bound. Whether a feature is causally emergent or decoupled in the paper’s formal sense (Un⁽ᵏ⁾ > 0) depends on the decomposition chosen. NOTE-987 shows that Ψ > 0 does not establish it for the candidate feature under the minimum-MI decomposition. So the parity example shows that Ψ’s test for decoupling passes a dependent feature, not that formal decoupling and independence come apart in every decomposition.
• Not for noisy features as written. For V_t = F(X_t) + ν_t, step 1 needs I(V_t; V_t′ | X_t) = 0. That holds when the noise at t is independent of everything at t′. The paper’s Definition 1 does not state it in that form.
• Not the multi-lag dependence. T₁ conditions on V_t alone. Barnett and Seth’s dependence conditions on V’s history and is smaller by I(V_t′; V_past | V_t), the memory of the reduced process. Ψ penalizes the larger, one-step quantity, so a feature that is independent only given its history is still penalized.
• Not that Ψ > 0 marks interaction or integration. The AR(1) and copying examples pass with independent parts. Ψ > 0 marks a feature whose own persistence beats the sum of what each part says about its future.
• Not a ranking of levels. Neither Ψ nor T₁ prices the size of the feature. W can grow with the number of parts summed, as in the AR(1) case.

Connections

• THEORY-257, THEORY-733. This places the ΦID practical criterion relative to dynamical independence. On closure the two agree in sign: Ψ rewards it. They differ on what else counts. Ψ adds the parts’ whole-minus-sum and penalizes redundancy among them. Independence asks for nothing beyond closure, so a lone self-copying part qualifies. In a linear Gaussian VAR(1), THEORY-733’s invariant subspaces have T₁ = 0, so for them Ψ reduces to W.
• THEORY-497 holds the ΦID account this criterion serves. The examples here show its practical test detects persistence of aggregates of independent parts.
• QUESTION-115: answers in part.
  • Where dynamical independence and the ΦID criterion part: a single persistent part is independent and not emergent, and a parity whose persistence depends on one part passes Ψ’s decoupling test and is not independent.
  • What each tracks: independence tracks closure. Ψ tracks closure plus collective, non-redundant predictability.
  • Neither tracks interaction, and neither prices size.
• THEORY-222. Effective information ranks levels by how much they say under interventions. Ψ cannot, since I(V_t; V_t′) ≤ I(X_t; X_t′). The two are not rivals on one scale.
• CLAIM-388. The AR(1) sum is the uncoupled case (coupling 0) of its uniformly coupled VAR(1), where the sum of all units is exactly independent. Here it also passes Ψ.