Contribution
It extends causal feature learning from a cause of a given target behaviour (UAI 2015) to the case where both the macro cause and the macro effect are built from micro data. It defines the pair, gives an algorithm that learns it from experiments, proves the coarsening theorem for both sides, and defines coarser subsidiary cause–effect pairs, which can compose into the fundamental one.
Key insight
Cluster the rows and the columns of the interventional matrix P(j | man(i)) at the same time. Rows that agree are one macro cause; columns that agree are one macro effect. Structure in J that does not depend on I, such as the distractor neurons’ rhythm, is ignored because it does not change any column’s profile across manipulations, where simply clustering J would split the true classes in half.
Assumptions
- A causal ml-system: discrete I and J of possibly huge cardinality, a hidden confounder H, P(J, I) = Σ_H P(J | I, H)P(I | H)P(H), with independent noise on each side.
- A micro manipulation man(I = i) sets I without touching H or J: P(J | man(i)) = Σ_H P(J | i, H)P(H) (Def. 1).
- No distribution over interventions: “unless we assume a distribution P(man(I)) over the interventions, we cannot apply the notion of a sufficient statistic to manipulations in the I space. Following Pearl’s approach, we refrain from specifying intervention distributions” (§2.2).
- For Algorithm 1, P(j | man(i)) smooth in i and j; for the theorem, Lebesgue measure on the free parameters.
- Micro variables “do not lump together atoms that belong to different ‘true fundamental partition’ cells” (§5).
Key results
- Fundamental causal partition (Def. 2). i₁ ~ i₂ iff P(j | man(i₁)) = P(j | man(i₂)) for all j; j₁ ~ j₂ iff P(j₁ | man(i)) = P(j₂ | man(i)) for all i. Their labels are the fundamental cause C and effect E (Def. 3); do(C = c) is man(i) for any i with C(i) = c (Def. 4).
- Sufficient causal description (Theorems 5, 9, 12). E is “the minimal sufficient statistic for P(J | man(i)) for any i” among partitions of J (Fisher factorization); C and E “losslessly recover P(j | man(i))”, and any other partition “is either finer than C, E or does not define unambiguous manipulations”. The authors add that E is minimal only among partitions; a histogram is a smaller sufficient statistic.
- Fundamental causal coarsening (Theorems 7, 13). Among generative distributions with given observational partitions of I and J, those whose causal partition of I, or of J, is not a coarsening of the observational one have measure zero. Part (1) is referred to the UAI 2015 proof; part (2) writes P(j₁ | i) = P(j₂ | i) as Σ_h [α_{j₁,h,i} − α_{j₂,h,i}] β_{h,i} γ_h = 0, a polynomial in β = P(I | H) that is not identically zero, then integrates by Fubini.
- Subsidiary variables (Def. 8, Alg. 2). Strict coarsenings (C̄, Ē) with P(Ē | do(C = c)) constant over the C-cells inside each C̄-cell and distinct across C̄-cells. “v-bar” is not a cause of “pulse”: an image with a v-bar and no h-bar gives P(pulse | man(i)) = 0, one with both gives 0.8, an ambiguous manipulation. Non-interacting subsidiary pairs factor the fundamental table (Def. 11); “in almost all the cases” there are none.
- Simulation. 10,000 images in four classes, 100 Izhikevich neurons, P(pulse | do(h-bar)) = P(30 Hz | do(v-bar)) = 0.8; Algorithm 1 recovers the four-by-four table, with clusters merged by a KL threshold, and “the best solution in practice is to cluster them by hand”.
Claims
| id | claim | strength | support |
|---|---|---|---|
| C1 | (C, E) is the coarsest partition pair that losslessly recovers P(j | man(i)) | moderate: right for C, wrong for E as stated | Theorem 12 proof; fails for proportional columns (below) |
| C2 | The causal partitions generically coarsen the observational ones on both sides | strong, as genericity | Theorem 13 with proof; slips in (2ii) do not break it |
| C3 | Algorithm 1 recovers the fundamental cause and effect | weak in general | one simulation built with that structure |
| C4 | Non-interacting subsidiary causes indicate independent mechanisms | weak | asserted |
| C5 | The interventional/observational distinction is “entirely lost” in Hoel et al. (2013) | weak | one sentence, no computation |
| C6 | Whether macro causes exist is an empirical question, since they “cannot be defined arbitrarily” | moderate | argument (§5) |
Concepts
- Fundamental cause and effect: the class labels of the row and column partitions of P(j | man(i)).
- Subsidiary cause–effect pair: a coarsening of both that keeps manipulations unambiguous for the coarser effect.
- Partition product: the coarsest common refinement; composition of macro variables (Def. 10).
Connections
It generalizes Chalupka, Perona and Eberhardt 2015 (UAI, not registered). Its overview is LIT-1834 (NOTE-1415). Its cause partition is the optimum of the causal information bottleneck (LIT-1802, THEORY-1125). It credits computational mechanics (LIT-1126) and Shalizi. Eberhardt and Lee (LIT-1839) recommend it against Hoel’s effective information (LIT-1038), which it dismisses in one sentence. The effect-side coarsening it defines is the one Hoel’s coarse-grainings of a transition matrix also make.
Bearing on the record
- QUESTION-204, second half. This is the paper that supplies the test the question names, for cause and effect together. It is independent of EI-type choices by design: the authors decline to put any distribution on interventions, and the partition is defined from point manipulations alone. THEORY-1169, filed from this reading, shows that EI under one common full-support distribution ties the micro exactly at this partition (with effects grouped by normalized likelihood) and its refinements, whatever the distribution. On which coarse-grainings lose nothing, the causal criterion of THEORY-1167’s set-up and the interventionist test agree. Hoel’s uniform-per-level EI does not: on Eberhardt and Lee’s (13) and (14) it merges causes with different rows, and on their note 6 it keeps identical rows apart. The paper computes nothing on Hoel’s systems; that comparison is the record’s.
- THEORY-787. The verdict CFL gives on a transition matrix is a partition, not a level where causation is stronger: the micro description “contains no more causal information”, but it loses none either. Where micro rows are exactly equal, as in Hoel’s (11), CFL and EI pick the same macro; elsewhere CFL keeps the micro. So CFL can confirm that a macro is proportionate, never that it beats the micro. Hoel 2013’s own examples (THEORY-222) were not recomputed.
- QUESTION-115. Its row and column clustering finds a cause and an effect with different partitions. On systems where EI’s single partition and CFL’s pair differ, which level each criterion picks can be read off at once (note 6).
- QUESTION-252. The sufficient-description theorem is a predictive sufficiency condition for a declared target under intervention, and the coarsening theorem relates it to observational sufficiency: generically the interventional condition holds wherever the observational one does, never the reverse under confounding. Subsidiary variables are coarser descriptions that keep sufficiency for a coarser declared target only.
Limitations
- The minimality claim is wrong as stated. Definition 2 groups effects by equal columns. Two effect states with proportional, unequal columns, T(j₁ | i) = κ T(j₂ | i) for all i, can be merged with no loss: given the merged count, the split is binomial with a parameter that does not depend on i, so the merged partition is sufficient for the family {P(J | man(i))}. Example (record script): rows (0.2, 0.4, 0.4) and (0.1, 0.2, 0.7); merging the first two effects keeps I(X; Y) exactly (0.0667 bits under uniform w, 0.0238 under (0.9, 0.1)). The proof treats each p_j as a free parameter, not as a function of i, which is where it goes wrong. The overview’s §3.3 normalizes likelihood vectors before merging, which is the right criterion, but for clusters only.
- Proof slips in Theorem 13 (2ii). “Set β_{h*,i} = 1 (and thus β_{h,i} = 0 for any h ≠ h*)” does not follow, since β_{h,i} = P(i | h) is normalized over i, not h; both can be set, so the argument survives. The conclusion then writes ”≠” where ”=” is meant.
- PMLR’s metadata orders the authors Chalupka, Eberhardt, Perona; the PDFs give Chalupka, Perona, Eberhardt. The arXiv PDF prints “JMLR: W&CP volume 41”; the volume is PMLR 51.
- One simulation, built with the structure recovered; merge thresholds chosen by hand.
- Exact equality throughout: with estimated rows, the partition is a threshold choice.
Open questions
- With the effect classes corrected to normalized columns, does the coarsening theorem’s part (2) still hold as stated? Its constraint is written for equal columns.
- What happens “when this assumption is violated”, that micro variables do not straddle true cells, is left open (§5).