nucleation

ARG-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

4 min read

Properties7
StatusProposed
Formformal-application
Tagscausality, information-theory, philosophy-of-science
Lineformal-apparatus
GroundsTHEORY-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, THEORY-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, LIT-1038 — Quantifying causal emergence shows that macro can beat micro, LIT-1839 — Causal Emergence: When Distortions in a Map Obscure the Territory, LIT-1843 — Causal emergence is widespread across measures of causation, LIT-1833 — Multi-Level Cause-Effect Systems
ConcludesCLAIM-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
Rendered intodocs/arguments-by-conclusion.md

Graph View

The inference

  1. Where the macro advantage comes from. LIT-1038 scores each level with its own uniform distribution over interventions, and finds macro descriptions with more effective information than the micro (THEORY-222). Score both levels with one distribution over micro states, and no coarse-graining of a transition exceeds the micro (THEORY-1167, by data processing). So the advantage is the gain from changing the distribution. Comolatti and Hoel concede the stationary case (LIT-1843).
  2. Why that gain is not the system’s. The distribution is chosen by the analyst. Eberhardt and Lee call the maximum-entropy choice “extraneous to the system” (LIT-1839). A quantity that changes sign when the analyst changes the distribution, with the mechanism held fixed, is not a property of the mechanism. THEORY-787 had tied itself to exactly this test (“in trouble if the macro advantage disappears whenever the same intervention distribution over micro states is used at both levels”). So it is rejected on its own terms.
  3. What survives as the system’s. The coarse-grainings that keep all of the micro effective information are the same for every full-support distribution (THEORY-1169). They merge only causes with identical interventional rows, and only effects with proportional columns. So which coarse-grainings lose nothing does not depend on the analyst’s choice. They are also the partitions that causal feature learning reaches by a criterion that uses no effective information and no distribution: two micro states share a macro state iff intervening to either gives the same effects (LIT-1833). Two independent routes reach the same partitions.
  4. The bridging premise. Among the descriptions that lose no causal information, the causes are found at the coarsest: a finer one draws distinctions that no intervention tells apart, and a coarser one loses some. This is proportionality. It is not derived here. It is the step from “what loses nothing” to “where the causes are”, and it is the record’s own.
  5. Conclusion. A macro description can be where a system’s causes are, as the proportionate one, but never by carrying more causal information than the micro (CLAIM-497).

Steps 1 to 3 are demonstrated: they are theorems the record checked numerically. Step 4 is interpretive, and the conclusion inherits its warrant.

Critical questions

  • Does the mapping premise hold? The results concern effective information of a known transition matrix. The question is where a system’s causes are. The mapping takes “causal information” to be effective information under some distribution with full support. That is LIT-1038’s own measure, so the application meets THEORY-787 on its own ground. It does not cover measures that data processing does not bound, such as per-transition or normalized ones (LIT-1843). The claim says so.
  • Are the assumptions met? Each has a scope limit:
    • A discrete, known mechanism. Estimated systems need a threshold for “the same effect”. That is what CLAIM-497’s defeated_if tests.
    • One transition. Macros spanning several micro steps are not covered; that case is in QUESTION-335.
    • Full support. A distribution that never intervenes on some micro state cannot tell it apart from anything. Then the ties can be coarser than the causal partition.
  • Is step 4 the only reading? No. Two rivals would stop short of it:
    • The micro is always the causal level, and coarse descriptions are only convenient. Step 4 answers that the extra micro distinctions make no difference to any intervention, so it is the micro that is disproportionate. Whether that answer is decisive is a question for the philosophy of causation. Yablo’s proportionality is the standard statement of it, and it is named, not registered.
    • Several levels are equally causal. This holds whenever a finer partition is also lossless. Step 4 picks the coarsest, which is unique. “Same interventional effects” is an equivalence relation, so its partition is the coarsest lossless one, and every lossless partition refines it (THEORY-1169).
  • Would more evidence discriminate? On estimated systems, yes: whether a proportionate level is stable across thresholds and sample sizes, which is the claim’s defeated_if. On toy matrices, no: there it holds by construction.

Backlinks