Source
Chalupka, Perona and Eberhardt (2015; AISTATS 2016), LIT-1833, read in NOTE-1420: Definition 2, the fundamental causal partition, i₁ ~ i₂ ⇔ P(j | man(i₁)) = P(j | man(i₂)) for all j, and j₁ ~ j₂ ⇔ P(j₁ | man(i)) = P(j₂ | man(i)) for all i; “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). Their overview, LIT-1834, read in NOTE-1415, §3.3: merging effect clusters after “Normalizing the likelihood vectors such that they sum to 1”. Effective information under a common distribution is THEORY-1167’s, from Eberhardt and Lee, LIT-1839, and Hoel et al., LIT-1038. The equivalence below is the record’s.
What was actually shown
Set-up. Finite cause states X and effect states Y (they may be one state set at two times). T(y | x) = P(Y = y | do(X = x)). An intervention distribution w over X with w(x) > 0 for every x, fixed in advance. EI_w = I(X; Y) with X ~ w, Y ~ T(· | X). A coarse-graining is a pair of partitions, f of X and g of Y. As in THEORY-1167, the macro cause is given its image law under w and each do(f(X) = u) is realized by w within the block, so the macro EI is I(f(X); g(Y)) under the same w.
The loss splits in two. Since g(Y) is a function of Y and f(X) of X,
I(X; Y) − I(f(X); g(Y)) = I(X; Y | g(Y)) + I(X; g(Y) | f(X)).
Both terms are non-negative, which is THEORY-1167’s bound again.
Effect side. I(X; Y | g(Y)) = 0 iff X is independent of Y given g(Y), i.e. w(x)T(y | x)/P(y) is the same for every y in a block. With w of full support, that is: within each block of g, the columns T(y | ·) are proportional, T(y | x) = κ_{y,y′} T(y′ | x) for all x. The condition does not mention w, so it holds for one full-support w iff it holds for all. (A state no cause reaches has a zero column and may go in any block.)
Cause side. I(X; g(Y) | f(X)) = 0 iff, within each block of f, the rows P(g(Y) | x) are equal. When the effect side holds, T(y | x) = c_y · P(g(Y) = g(y) | x) with c_y independent of x, so equal macro rows mean equal micro rows. Hence the whole loss is zero iff f merges only causes with identical rows of T and g merges only effects with proportional columns, whatever full-support w is used.
The coarsest tie. The coarsest f is the partition of X by T(· | x): Chalupka et al.’s fundamental cause. The coarsest g is the partition of Y by normalized columns. It equals their fundamental effect (equal columns) unless two effect states have proportional, unequal columns, in which case it is coarser and still loses nothing (NOTE-1420 records the example rows (0.2, 0.4, 0.4) and (0.1, 0.2, 0.7), whose first two effects merge with zero loss under every w checked). Every merge of two causes with different effects, an ambiguous manipulation in Spirtes and Scheines’s sense, loses information under every full-support w.
On Eberhardt and Lee’s matrices (computed, NOTE-1413). (11): rows 1–7 are identical, so the causal partition is {1–7}, {8}, the same as EI’s choice. (13): all eight rows differ, so the causal partition is the micro, and EI’s {1–7}, {8} is a lossy merge. (14) and its variant: all rows differ; EI merges x₁ and x₂ at a loss of 0.074 bits under w_H. Note 6: rows x₂ and x₃ are identical, so the causal partition merges them, which EI does not do, because Hoel coarsens cause and effect by one partition and the columns of y₂ and y₃ are not proportional. (17): no lossless merge.
What could have come out the other way: a coarse-graining with zero loss that merges two different rows, or two non-proportional reachable columns, under some full-support w. The search found none, and the identities exclude it.
What this does not say
- Not that the macro is where causation is. The micro always ties. The causal partition is the coarsest description that loses nothing, a proportionality notion in Yablo’s sense, not a level with more causation. THEORY-787’s claim that the macro is causally privileged gets no support from it.
- Not a ranking of lossy coarse-grainings. Whether a loss is zero does not depend on w; its size, w-weighted divergence among the merged rows, does. Any trade of loss against size, Hoel’s included, brings a distribution back.
- Not for approximate equality. With noisy or estimated rows, no two are equal and only the micro ties. Chalupka et al.’s algorithms merge by a threshold, KL or earth mover’s distance, which is a choice the theory does not make.
- Not for Hoel 2013’s temporal macros, continuous variables, or partitions forced to be the same on cause and effect. When f must equal g, the coarsest tie is the coarsest partition meeting both conditions at once, which can be the micro when f alone could merge (note 6).
- Not about confounding. T is interventional. Getting the causal partition from observational data needs Chalupka et al.’s coarsening theorem, a genericity result.
- The cause side alone, for a different information measure (causal information gain under a protocol), is THEORY-1125’s.