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THEORY-1167: When one intervention distribution over micro states is used at both levels, the effective information of a coarse-graining of a discrete mechanism, with each macro intervention realized by that distribution within its block, cannot exceed the micro effective information; so a positive causal emergence of effective information, scored with a uniform distribution at each level, equals the micro gain from changing the distribution minus a non-negative loss, and is a gain from the change of distribution

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Properties9
StatusActive
Version2
Tagscausality, information-theory, complex-systems
SourceLIT-1843 — Causal emergence is widespread across measures of causation, LIT-1839 — Causal Emergence: When Distortions in a Map Obscure the Territory, LIT-1038 — Quantifying causal emergence shows that macro can beat micro
ExtendsTHEORY-222 — With uniform interventions imposed separately at each level, a coarse-grained macro description of a known discrete mechanism can have more effective information than the micro description it supervenes on, so the level at which effective information is highest can be a macro level
CorrectsTHEORY-787 — A system's causal structure can be most effective at a macro scale, so that causation emerges at a coarse-grained level and the macro description is the one at which the system's causes are found, as Hoel, Albantakis and Tononi propose from effective information
Bears onQUESTION-204 — Does a macro description's effective-information advantage survive when the same intervention distribution over micro states is used at both levels, and does the EI-optimal level agree with an independent interventionist test of which level the effects track?
Cited as “extends” byTHEORY-1169 — For a discrete mechanism scored with one full-support intervention distribution over micro causes at both levels, a coarse-graining of cause and effect keeps all of the micro effective information if and only if it merges only causes with identical rows P(Y | do(x)) and only effects with proportional columns, whatever the distribution; so the coarsest coarse-graining that ties the micro is the causal partition of causal feature learning, with effects grouped by normalized likelihood, and any merge of causes with different effects loses information under every such distribution
Cited as “grounds” byARG-041 — From the effective-information bound and its ties to proportionality: a gain that depends on the intervention distribution is not a property of the system, the coarse-grainings that lose nothing are, and the coarsest of them is where the causes are, CLAIM-497 — A system's causes are found at the coarsest description whose interventions are unambiguous: a macro description is causally proportionate where it merges only micro states that intervention cannot tell apart, and no coarse-graining carries more causal information than the micro description it supervenes on

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Source

Comolatti and Hoel (arXiv 2022), LIT-1843, read in NOTE-1416: “the only condition to not show causal emergence was the overall effective information when using the observational distribution. This was known [Hoel 2017]”. Eberhardt and Lee (2022), LIT-1839, read in NOTE-1413, trace EI’s ambiguity and non-commutativity to the maximum-entropy distribution “extraneous to the system”. EI and its macro interventions are Hoel, Albantakis and Tononi’s (2013), LIT-1038, THEORY-222. The general bound and the decomposition below are the record’s.

What was actually shown

Set-up. A mechanism on a finite state set with transition matrix T(y | x), read as P(Y = y | do(X = x)). A partition f of the states, applied to cause and effect. An intervention distribution w over micro states, fixed in advance. EI under w is I(X; Y) with X ~ w and Y ~ T(· | X).

Common distribution. “The same distribution at both levels” means that the macro cause U = f(X) has the image law w(f⁻¹(u)), and the intervention do(U = u) is realized by drawing x from w(· | u). Then the macro transition matrix is

P(V = v | do(U = u)) = Σ_{x ∈ u} w(x | u) Σ_{y ∈ v} T(y | x),

which is P(f(Y) = v | f(X) = u) under X ~ w. So EI(macro) = I(f(X); f(Y)).

The bound. f(X) — X — Y — f(Y) is a Markov chain, so by data processing I(f(X); f(Y)) ≤ I(X; f(Y)) ≤ I(X; Y). So EI(macro) ≤ EI(micro) for every partition and every common w. Taking w to be the stationary distribution gives Comolatti and Hoel’s concession. Taking w uniform with blocks of equal size gives their symmetric bipartite model, where the macro’s 1 bit is below the micro’s at every setting computed (1.043 to 3.326 bits; exactly 1 at full degeneracy). A macro can at most tie.

Where Hoel’s gain comes from. Hoel’s macro EI uses the uniform distribution over macro states, each realized uniformly within its block. That is the common distribution w_H(x) = 1/(|U| · |block of x|), which is the micro uniform u only when all blocks have one size. So

CE = EI_macro(w_H) − EI_micro(u) = [EI_macro(w_H) − EI_micro(w_H)] + [EI_micro(w_H) − EI_micro(u)].

The first bracket is ≤ 0 by the bound. So CE > 0 requires the micro mechanism to carry more EI under w_H than under the uniform, and CE is at most that gain. On Eberhardt and Lee’s matrices: (11), CE 0.456 = 0 + 0.456; (13), CE 0.194 = −0.150 + 0.344; (14), CE 0.0001 = −0.074 + 0.074 (bits, computed).

What could have come out the other way: a single partition and distribution with macro EI above micro EI would refute it, and the random search found none.

What this does not say

  • Not that macro descriptions are useless or that coarse-graining never helps. Measures not bounded by data processing, such as per-transition effect information and quantities normalized by log₂ n, can rise at a macro under a common distribution (NOTE-1416). The bound is about EI, which is the measure THEORY-787 rests on.
  • Not the second half of QUESTION-204, whether the level EI picks agrees with an independent interventionist test. Eberhardt and Lee’s (14) is one hand-built case where it does not (NOTE-1413).
  • Not Hoel 2013’s temporal or spatio-temporal macros, where the macro effect is several micro steps ahead and the micro comparison is with one step. The bound covers a coarse-graining of one transition, with any partitions of cause and effect. The four examples behind THEORY-222’s 0.40 to 2.92 bits were not recomputed.
  • Not that the uniform distribution is wrong. It says what the gain is a gain of. Whether an analyst’s distribution should differ between levels is the interpretive question THEORY-787 takes a side on.

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