Contribution
A critique of Hoel’s 2017 account of causal emergence on two desiderata any account of multi-scale causation should meet: macro variables without (much) ambiguity of manipulation, and abstraction that commutes with marginalization over time. It gives numerical examples where EI maximization fails each, and locates the cause in the maximum-entropy intervention distribution and the averaging steps.
Key insight
EI is mutual information under an input distribution the analyst supplies. A uniform distribution over macro states, each realized as a uniform average of its micro states, is a non-uniform distribution over micro states. So which coarse-graining wins depends on that injected distribution, not only on P(Y | do(X)). An account built on the interventional conditionals alone, as in Chalupka et al.’s causal feature learning, avoids both failures.
Assumptions
- A discrete micro state space with a transition matrix read as P(Y | do(X)): no hidden variables, one-step transitions. No marginal P(X) is given.
- Hoel’s set-up: cause and effect share a state space and are coarsened together; EI(X → Y) = I(X; Y) with X uniform (eq. 7); a macro intervention do(U = u) is the uniform average of do(X = x) over x ∈ u (eq. 8); causal emergence is a strict coarsening maximizing EI, the maximum being the causal capacity (eq. 10).
- Distances between rows (absolute, squared Euclidean, max and min component difference, symmetrized KL) to say how different two merged states are.
Key results
- Hoel’s examples. (11): micro EI “0.55”, macro {1–7, 8} EI 1. The record’s computation gives 0.544 for the micro. (13): micro EI 0.81 (checked, 0.806), macro 1; rows sum to 1.
- Ambiguity (§3). The 3×3 matrix (14), rows (0.7, 0.1, 0.2), (0.4, 0.5, 0.1), (0, 0, 1), has micro EI 0.6787 and is maximized by merging x₁ and x₂, macro EI 0.6788 (checked: 0.67867 against 0.67879), though the merged rows are at absolute distance 0.8. Changing the first row to (0.8, 0.1, 0.1) keeps the micro (checked: 0.761 against 0.758), though x₁ and x₂ still look more alike than either does like x₃. Appendix A lists matrices on the 0.1 grid where the merged rows are far apart under each distance.
- Non-commutativity (§4). For (17), rows (0.3, 0.2, 0.5), (0.4, 0.1, 0.5), (0.5, 0.5, 0): A(T²) keeps all three states (EI 0.0524), while A(T)² is the 2-state [[0.75, 0.25], [0.5, 0.5]] (EI 0.0488). Checked exactly. “Only very specific transition probability matrices satisfy commutativity”, which is asserted, not shown.
- Note 6. With rows (0, 0.2, 0.8), (0, 0, 1), (0, 0, 1), EI keeps x₂ and x₃ apart although their rows are identical (checked: 0.113 against 0.052), because cause and effect must be coarsened together.
- §5. Causal capacity is like channel capacity, a normative quantity about an optimal input that the system need not use, so Hoel’s is an account of potential, not actual, emergence. The EI-maximizing partition need not be unique; two incomparable maximizing partitions are conjectured, not found (note 11).
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | EI maximization can merge micro states with very different effects | strong | (14) and Appendix A, numerical, checked |
| C2 | The EI-optimal abstraction does not commute with two-step marginalization | strong for the example | (17), checked |
| C3 | Commutativity fails for most matrices | weak | asserted, no count or proof |
| C4 | Both failures come from the maximum-entropy input distribution | moderate | argument; no computation under any other distribution |
| C5 | Causal capacity is normative: it describes potential, not actual, emergence | moderate | analogy with channel capacity and Hoel’s own passage |
| C6 | Two incomparable EI-maximizing partitions can exist | weak | conjecture (note 11) |
Concepts
- Ambiguous manipulation (Spirtes and Scheines): an intervention on a macro variable whose effect depends on which micro realization is set, as total cholesterol with HDL and LDL.
- Proportional cause (Yablo): the coarsest description of the cause that screens off finer ones from the effect, i.e. all of whose interventions are unambiguous.
- Commutativity of abstraction and marginalization: A(T × T) = A(T) × A(T), eq. (16).
- Causal capacity: max EI over coarsenings, eq. (10).
Connections
It criticizes Hoel 2017 (“When the map is better than the territory”, not registered), the successor of Hoel, Albantakis and Tononi 2013 (LIT-1038, THEORY-222). The commutativity demand is credited to Rubenstein et al. (LIT-1334) and Beckers and Halpern (LIT-1316), whose consistency conditions are stronger; the alternative it recommends is Chalupka, Perona and Eberhardt’s causal feature learning (not registered). It cites Rosas et al.’s “Reconciling emergences” (LIT-1244) and Aaronson’s blog post for similar concerns, and Dewhurst 2021 for the epistemic reading. The defenders’ replies the record holds are CE 2.0 (LIT-1136, NOTE-1387), which answers the uniform-intervention criticism, and Zhang et al. (LIT-1129, NOTE-1399), which answers ambiguity and non-commutativity with a 3×3 example. Comolatti and Hoel 2022 (LIT-1843) answers the same criticism by moving to other measures.
Bearing on the record
- THEORY-787 and QUESTION-204. The triage expected the published case that the EI advantage is an artefact of the uniform choice. The paper argues that, and shows two consequences of the choice, but it does not run the test QUESTION-204 asks for. Nowhere does it compute EI with one distribution over micro states at both levels. That test needs no example: under any single micro distribution, the macro EI is I(f(X); f(Y)) ≤ I(X; Y) by data processing, so the advantage cannot survive (THEORY-1167). The paper supplies the diagnosis and the record’s derivation supplies the proof. Together they meet THEORY-787’s “in trouble if” clause for EI.
- QUESTION-204, second half. Example (14) is a small case of the disagreement the question asks about. An interventionist test of realization insensitivity (Yablo’s proportionality, THEORY-133’s) would keep x₁ and x₂ apart. Their effects on the macro effect variable still differ (0.8/0.2 against 0.9/0.1), and EI merges them. It is a hand-built 3×3 matrix, so by the question’s own terms it cannot settle agreement in general. It does show that the EI-optimal level and a proportionality test can come apart.
- The paper’s own examples, corrected. (13)‘s merged states are ambiguous only for the micro effect Y. Every one of states 1–7 sends all its mass into the block {1–7}, so for the macro effect V, the variable it is scored on, the manipulation is unambiguous. The partition is strongly lumpable. The ambiguity charge stands on (14), not on (13). The commutativity example is the same kind of case: {x₁, x₂} is a strongly lumpable partition of (17), since both rows send 0.5 into each block. A(T)² equals the {x₁, x₂} coarsening of T², the same matrix [[0.75, 0.25], [0.5, 0.5]]. So the coarse-graining commutes with time evolution, as THEORY-822’s lumpability guarantees, and only the EI argmax changes, from that partition at one step to the micro at two. The failure is in the selection criterion, not in the coarse-graining.
- QUESTION-115 (A21). A21’s condition, C_*(P_ℓ^{do(a)}) = P_L^{do(ā)} for a declared family of interventions, is this paper’s commutativity and ambiguity demands in one line. The paper shows that EI’s family, uniform averages over realizers, passes coarse-grainings that fail both. The same answer for EI as the ΦID note of 2026-10-20 gives for Ψ: the criterion picks a level the admissibility condition does not license.
- QUESTION-252. One more pairwise separation: strong lumpability (D and M hold for the block label) does not make the EI-optimal choice stable under a change of time step. That is a property of EI, not one of the question’s five conditions, so it is context, not a witness.
No instruction for machine-learning practice; nothing for the anthology.
Limitations
- The examples are found by a grid search at step 0.1 on 3×3 matrices, with no analytic result. The commutativity claim for “most” matrices is not quantified.
- The paper never computes EI under a distribution other than the maximum-entropy one. So C4 is argued, not shown, and the paper does not check whether its two failures survive another distribution.
- It reads Hoel 2017. Hoel 2013’s spatial and temporal macros, and CE 2.0, are not addressed.
- Errors and slips. (11)‘s micro EI is printed 0.55 where 0.544 is computed. Total cholesterol’s components are written “LDP” once. Hoel’s eq. (7) is criticized as notation, which is fair, and otherwise the restatements check.
Open questions
- Under the system’s own stationary distribution used at both levels, does a partition ever tie with the micro on EI without being a sufficient statistic of the cause for the coarse effect? A tie is the most a macro can get under a common distribution (THEORY-1167).
- Does a fixed, system-given intervention family exist on which EI’s level and a proportionality test can be compared on a system not built for it?
Note of 2026-10-29: causal feature learning, registered and read
Connections calls the alternative this paper recommends, Chalupka, Perona and Eberhardt’s causal feature learning, not registered. That is out of date. It is registered and read: the AISTATS 2016 paper, LIT-1833 (NOTE-1420), and the overview, LIT-1834 (NOTE-1415). The record’s THEORY-1169, filed from those readings, takes the step this paper leaves undone. Under one full-support distribution at both levels, a coarse-graining keeps all of the micro EI iff it merges only causes with identical rows and only effects with proportional columns, which are causal feature learning’s partitions. On this paper’s matrices: (11)‘s causal partition is EI’s choice, {1–7}, {8}; in (13) and (14) all rows differ, so the causal partition is the micro and EI’s merges are lossy; note 6’s identical rows are merged by the causal partition and kept apart by EI. So the paper’s recommendation is borne out on which coarse-grainings lose nothing. Hoel 2017, also called not registered, still is not.